aberrations of anamorphic optical...

206
Aberrations of Anamorphic Optical Systems Item Type text; Electronic Dissertation Authors Yuan, Sheng Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 27/04/2018 06:44:25 Link to Item http://hdl.handle.net/10150/195267

Upload: vongoc

Post on 06-Feb-2018

245 views

Category:

Documents


8 download

TRANSCRIPT

Page 1: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

Aberrations of Anamorphic Optical Systems

Item Type text; Electronic Dissertation

Authors Yuan, Sheng

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.

Download date 27/04/2018 06:44:25

Link to Item http://hdl.handle.net/10150/195267

Page 2: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS

By

Sheng Yuan

A Dissertation Submitted to the Faculty of the

COLLEGE OF OPTICAL SCIENCES

In Partial Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

2008

Page 3: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

2

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Sheng Yuan entitled Aberrations of Anamorphic Optical Systems and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy _______________________________________________________________________ Date: 11/05/08

Jose M. Sasian, Dissertation Director _______________________________________________________________________ Date: 11/05/08 Tom Milster, Committee Member _______________________________________________________________________ Date: 11/05/08

Hong Hua, Committee Member _______________________________________________________________________ Date: _______________________________________________________________________ Date: Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ________________________________________________ Date: 11/05/08 Dissertation Director: Jose M. Sasian

Page 4: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

3

STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: Sheng Yuan

Page 5: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

4

ACKNOWLEDGMENTS

I am deeply grateful to my advisor, Dr. José Sasián, for supporting me in the study of this work. I appreciate your patient guidance which kept me on track throughout my thesis studies. I am also deeply grateful to the other members of my dissertation committee, Dr. Tom Milster and Dr. Hong Hua, for reviewing my thesis. Additional thanks go to Bruce Pixton, Peng Su and Lirong Wang for their help in editing my thesis. To my parents Zai-chao Yuan and Sie Wang, thank you for always encouraging and supporting me over the years of my study. Finally, to my lovely wife Grace Yuan, thank you for your unconditional love and support during my years of graduate study. Without you this thesis would not have been possible.

Page 6: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

5

TABLE OF CONTENTS

LIST OF FIGURES ............................................................................................................ 8

LIST OF TABLES............................................................................................................ 10

ABSTRACT...................................................................................................................... 11

CHAPTER 1 ..................................................................................................................... 12 INTRODUCTION ............................................................................................................ 12

1.1 What is an anamorphic system?.............................................................................. 13 1.2 Why is it a difficult problem? ................................................................................. 14 1.3 Historical background............................................................................................. 16 1.4 Dissertation content ................................................................................................ 18

CHAPTER 2 ..................................................................................................................... 20 FIRST-ORDER THEORY FOR ANAMORPHIC SYSTEMS........................................ 20

2.1 Direction cosines and their paraxial approximations.............................................. 20 2.2 Three dimensional ray transfer and refraction equations........................................ 22 2.3 Double curvature surface types and their surface normal—general theory............ 25 2.4 Double curvature surfaces and their surface normal—paraxial approximations.... 28 2.5 Ideal (first-order) image model for anamorphic imaging ....................................... 30 2.6 The paraxial ray tracing equations for anamorphic systems................................... 33 2.7 Paraxial image properties of anamorphic systems---part one................................. 35 2.8 Paraxial image properties of anamorphic systems---part two ................................ 37 2.9 Paraxial image properties of anamorphic systems---part three .............................. 40 2.10 Paraxial image properties of anamorphic systems---part four.............................. 43 2.11 Paraxial image properties of anamorphic systems---part five .............................. 48 2.12 Summary............................................................................................................... 49

CHAPTER 3 ..................................................................................................................... 51 GENERAL ABERRATION THEORY FOR ANAMORPHIC SYSTEMS .................... 51

3.1 Fermat’s principle and Sir Hamilton’s characteristic function............................... 51 3.2 Aberration function and ray aberrations for anamorphic systems.......................... 53 3.3 Power series expansion of aberration function ....................................................... 58 3.4 Summary................................................................................................................. 63

CHAPTER 4 ..................................................................................................................... 64 METHOD OF ANAMORPHIC PRIMARY ABERRATIONS CALCULATION.......... 64

4.1 The total ray aberration equations for anamorphic systems ................................... 64 4.2 Preparation for the anamorphic primary ray aberration equations deduction ........ 70 4.3 The anamorphic primary ray aberration calculation-part one ................................ 74 4.4 The anamorphic primary ray aberration calculation-part two ................................ 75 4.5 Summary................................................................................................................. 76

Page 7: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

6

TABLE OF CONTENTS-Continued CHAPTER 5 ..................................................................................................................... 77 PRIMARY ABERRATION THEORY FOR PARALLEL CYLINDRICAL ANAMORPHIC ATTACHMENT SYSTEMS ................................................................ 77

5.1 The primary ray aberration equations for cylindrical anamorphic systems............ 79 5.2 Primary ray aberration coefficients for parallel cylindrical anamorphic systems .. 81 5.3 Primary wave aberration coefficients for parallel cylindrical anamorphic systems86 5.3 Simplification of the results.................................................................................... 88 5.5 Summary................................................................................................................. 95

CHAPTER 6 ..................................................................................................................... 99 PRIMARY ABERRATION THEORY FOR CROSS CYLINDRICAL ANAMORPHIC SYSTEMS......................................................................................................................... 99

6.1 Primary ray aberration coefficients for cross cylindrical anamorphic systems ...... 99 6.2 Primary wave aberration coefficients for cross cylindrical anamorphic systems. 104 6.3 Simplification of the results.................................................................................. 106 6.4 Summary............................................................................................................... 111

CHAPTER 7 ................................................................................................................... 114 PRIMARY ABERRATION THEORY FOR TOROIDAL ANAMORPHIC SYSTEMS......................................................................................................................................... 114

7.1 The primary ray aberration coefficients for toroidal anamorphic systems........... 114 7.2 The primary wave aberration coefficients for toroidal anamorphic systems........ 119 7.3 Simplification of the results.................................................................................. 121 7.4 Summary............................................................................................................... 125

CHAPTER 8 ................................................................................................................... 127 PRIMARY ABERRATION THEORY FOR GENERAL ANAMORPHIC SYSTEMS127

8.1 The primary ray aberration coefficients for general anamorphic systems............ 127 8.2 The primary wave aberration coefficients for general anamorphic systems ........ 133 8.3 Simplification of the results.................................................................................. 135 8.4 Summary............................................................................................................... 139

CHAPTER 9 ................................................................................................................... 141 TESTING OF THE RESULTS....................................................................................... 141

9.1 The idea of data fitting.......................................................................................... 142 9.2 The detailed primary aberration coefficients data fitting steps............................. 144 9.3 A testing example ................................................................................................. 148 9.4 Summary............................................................................................................... 154

Page 8: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

7

TABLE OF CONTENTS-Continued CHAPTER 10 ................................................................................................................. 155 DESIGN EXAMPLES.................................................................................................... 155

10.1 An anamorphic singlet ........................................................................................ 156 10.2 An afocal anamorphic attachment ...................................................................... 161 10.3 An anamorphic Triplet design ............................................................................ 167

10.3.1 General methods for discovering an anamorphic beginning design............ 169 10.3.2 An anamorphic Triplet with an anamorphic ratio of 1.22 ........................... 171 10.3.3 Another anamorphic Triplet with an anamorphic ratio of 1.35 ................... 175

10.4 An anamorphic field lens design ........................................................................ 178 10.5 An anamorphic Double Gauss design with anamorphic ratio 1.5 ...................... 181 10.6 An anamorphic fisheye lens design with anamorphic ratio 3:4.......................... 188

10.6.1 An anamorphic fisheye design without field lens........................................ 190 10.6.2 An anamorphic fisheye design with field lens............................................. 194

10.7 Summary............................................................................................................. 197 CHAPTER 11 ................................................................................................................. 198 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK................................ 198

11.1 Conclusions......................................................................................................... 198 11.2 Suggestions of future work ................................................................................. 199

APPENDIX A................................................................................................................. 200 THE TYPICAL SHAPE OF ANAMORPHIC PRIMARY WAVE ABERRATIONS.. 200 REFERENCES ............................................................................................................... 203

Page 9: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

8

LIST OF FIGURES

Figure 1-1 Example of a double curvature surface………………………………………13

Figure 1-2 Example of an anamorphic system--a system made from cylindrical

lenses……………………………………………………………………………………..14

Figure 1-3 Constant astigmatism for a single double curvature surface………...……….15

Figure 2-1 Direction cosines of a ray………………………...……………………...…...21

Figure 2-2 Three dimensional ray tracing………………………………………………..23

Figure 2-3 Gaussian optics properties for the associated x-RSOS……...….……………36

Figure 2-4 The object and stop plane…………………………………………………….46

Figure 3-1 Wavefront error………………………………………………………………54

Figure 4-1 Refraction on surface j ……………………………………………………...66

Figure 5-1 A parallel cylindrical anamorphic attachment system……………………….81

Figure 5-2 The y-z (top) and x-z (bottom) symmetry planes of a parallel cylindrical

attachment system………………………………………………………………………..82

Figure 6-1 A cross cylindrical anamorphic system example…………………………….99

Figure 6-2 The two principal sections of a cross cylindrical anamorphic system……...100

Figure 9-1 Layout of a simple anamorphic system in the y-z (left) and x-z principal

sections………………………………………………………………………………….149

Figure 9-2 Grid distortion map…………………………………………………………150

Figure 10-1 Layout in the y-z (left) and x-z (right) principal sections…………………159

Figure 10-2 On-axis system performance………………………………………………159

Figure 10-3 Layout in the y-z (left) and x-z (right) principal sections…………………163

Figure 10-4 Layout in the y-z (left) and x-z (right) principal sections…………………164

Figure 10-5 Spot diagram………………………………………………………………164

Figure 10-6 Ray fan…………………………………………………………………….165

Figure 10-7 Layout in the y-z (left) and x-z (right) principal sections…………………173

Figure 10-8 Spot diagram………………………………………………………………173

Figure 10-9 Ray fan…………………………………………………………………….174

Figure 10-10 Layout in the y-z (left) and x-z (right) principal sections………………..176

Page 10: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

9

LIST OF FIGURES-Continued

Figure 10-11 Spot diagram……………………………………………………………..176

Figure 10-12 Ray fan…………………………………………………………………...177

Figure 10-13 Layout in the y-z (left) and x-z (right) principal sections………………..179

Figure 10-14 Spot diagram……………………………………………………………..180

Figure 10-15 Layout of the initial design in the y-z (left) and x-z (right) principal

sections………………………………………………………………………………….183

Figure 10-16 Spot diagram for the initial design……………………………………….184

Figure 10-17 Spot diagram and aberrations coefficients after the first stage optimization

…………………………………………………………………………………………..185

Figure 10-18 Layout of the anamorphic Double Gauss design………………………...186

Figure 10-19 Spot diagram……………………………………………………………..187

Figure 10-20 OPD fan…………………………………………………………………..187

Figure 10-21 A RSOS Double Gauss type fish-eye Lens………………………………189

Figure 10-22 Layout in the y-z (left) and x-z (right) principal sections………………..190

Figure 10-23 Grid distortion for the initial anamorphic fisheye design………………..191

Figure 10-24 Layout of the final design in the y-z (left) and x-z (right) principal

sections………………………………………………………………………………….192

Figure 10-25 Grid distortion map of the final design.………………………………….192

Figure 10-26 Spot diagram of the final design…………………………………………193

Figure 10-27 Ray fan of the final design……………………………………………….193

Figure 10-28 Layout of the final design in the y-z (left) and x-z (right) principal sections

…………………………………………………………………………………………..195

Figure 10-29 Vignetting plot…………………………………………………………...195

Figure 10-30 Spot diagram……………………………………………………………..196

Figure 10-31 Ray Fan…………………………………………………………………..196

Page 11: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

10

LIST OF TABLES

Table 9-1 Lens data……………………………………………………………………..150

Table 9-2 Full aperture is 10mm, HFOV is 10 degree…………………………………151

Table 9-3 Full Aperture is 4mm, HFOV is 4 degree…………………………………...152

Table 9-4 Full Aperture is 1mm, HFOV is 1 degree…………………………………...153

Table 10-1 Specifications of the anamorphic singlet…………………………………...159

Table 10-2 Lens data and remaining primary aberration coefficients………………….160

Table 10-3 Specifications of the anamorphic system…………………………………..164

Table 10-4 Lens data and remaining primary aberration coefficients………………….165

Table 10-5 Specifications of the anamorphic triplet with anamorphic ratio 1.22……...173

Table 10-6 Lens data and remaining primary aberration coefficients………………….174

Table 10-7 Specifications of the anamorphic triplet with anamorphic ratio 1.35……...176

Table 10-8 Lens data and remaining primary aberration coefficients………………….177

Table 10-9 Lens data and remaining primary aberration coefficients………………….180

Table 10-10 Specifications of the anamorphic Double Gauss………………………….183

Table 10-11 Lens data and remaining primary aberration coefficients………………...188

Table 10-12 Initial design specifications of the anamorphic Double Gauss …………..190

Table 10-13 Final design specifications of the anamorphic Double Gauss…………….191

Table 10-14 Lens data of the final design………………………………………………192

Table 10-15 Specifications of the anamorphic Double Gauss with field lens………….194

Table 10-16 Lens data of the final design with field lens………………………………195

Page 12: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

11

ABSTRACT

A detailed study of the aberrations of anamorphic optical systems is presented. This

study has been developed with a theoretical structure similar to that of rotationally

symmetric optical systems (RSOS) and can be considered a generalization.

A general method of deriving the monochromatic primary aberration coefficient

expressions for any anamorphic system types with double plane symmetry has been

provided.

The complete monochromatic primary aberration coefficient expressions for

cylindrical anamorphic systems, toroidal anamorphic systems and general anamorphic

systems with aspheric departure have been presented, in a form similar to the Seidel

aberrations of RSOS.

Some anamorphic image system design examples are provided that illustrate the use

and value of the theory developed.

Page 13: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

12

CHAPTER 1

INTRODUCTION

Optical systems can be classified into groups according to their symmetry [1], for

example, rotationally symmetric optical systems (RSOS) [2], double plane symmetric

optical systems (DPSOS) [3], and single plane symmetric optical systems (SPSOS) [4,5],

etc. Currently most optical systems are RSOS, and a tremendous amount of research is

going on in this field.

However, in recent years an interest in understanding DPSOS has arisen [6-9],

mainly because this type of system can offer a unique solution to anamorphic image

formation. For this reason, a DPSOS is also called an anamorphic system.

When optical designers want to use an anamorphic system, they will find that there

are not enough well established theoretical models and results as are found for RSOS,

such as the first-order theories and the Seidel third-order aberrations, etc. To date, several

papers and books have been published explaining some aspects of the behavior of

anamorphic systems [10-18], but most of these works lack enough theoretical structure,

and accordingly the results obtained are far from complete. The current situation on

anamorphic systems research can be summarized by the following statement —we do not

have the complete primary (third-order) aberration coefficients for anamorphic systems

that would provide necessary design insight, except for the simplest case [13].

In the work that is offered here, we will present a detailed study for the

monochromatic primary aberrations in anamorphic systems. This study will be developed

Page 14: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

13

with a theoretical structure similar to that of the RSOS. Even though the approach

method is different from the traditional wave aberration approach for RSOS, it can be

considered a generalization of RSOS.

This work will provide a clear understanding of anamorphic image formation with

first-order optics and the primary (third-order) aberration features for anamorphic

systems. And we will present the primary aberration coefficient expressions for the most

common types of anamorphic systems, in a form similar to the famous Seidel aberrations

of RSOS.

1.1 What is an anamorphic system?

An anamorphic system is an imaging system contains double curvature surfaces

which have two mutually perpendicular planes of symmetry. By a double curvature

surface we mean a surface which has different radii of curvature in two perpendicular

cross sections. Figure 1-1 shows an example of a double curvature surface.

Figure 1-1 Example of a double curvature surface

Page 15: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

14

By keeping the two symmetry planes mutually perpendicular to each other

throughout the system, an anamorphic system will have double plane symmetry. These

two symmetry planes are also called the principal sections of the anamorphic system.

Because the optical power is tied to the curvatures, an anamorphic system has

different optical powers in each principal section, thus forming an anamorphic image.

The effective focal lengths in both principal sections will determine the anamorphic ratio,

which is the ratio of the two magnifications, each associated with one principal section.

A common example of an anamorphic imaging system is a system made from cross

cylindrical lenses which can map a square object field into a rectangular image field.

Figure 1-2 shows an example of this configuration.

Figure 1-2 An example of anamorphic systems--a system made from cylindrical lenses

1.2 Why is it a difficult problem?

Anamorphic systems are different from RSOS from the first-order optics sense. If a

ray starts in one of the symmetry planes (principal sections) of an anamorphic system,

namely the x-z plane or the y-z plane, it will always stay in this plane as it is traced

through the system. For any ray not lying in one of the symmetry planes, it will be a skew

ray and will not be contained in any single plane as it is traced through the system.

Page 16: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

15

Due to the double curvature nature of the elements in an anamorphic system, we do

not have unique object and image points in each intermediate space. Instead, we have one

set of intermediate object and image points associated with the x-z symmetry plane and

another set of points associated with the y-z symmetry plane. Thus, a single double

curvature surface is not an imaging system because it can not form a single image point

for an on-axis object point. As a result, we intrinsically have constant astigmatism for any

single double curvature surface in the system, regardless of where we might locate our

observation plane (see Figure 1-3 below).

Figure 1-3 Constant astigmatism for a single double curvature surface

Even more, we have two sets of intermediate paraxial object and image planes

floating in space, each associated with one symmetry plane. In the final image space, we

will let the two image planes coincide with each other to complete the imaging formation.

In other words, an anamorphic imaging system will generally be constrained to have

Page 17: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

16

unique object and image planes in the object and final image space, but will not in the

intermediate spaces.

The same is true for pupils—generally we do not have a unique entrance pupil and

exit pupil in every intermediate space, except in the space where the stop is located.

Instead, we will have one set of unique pupils for each symmetry plane.

Due to these features, when we discuss optical path difference error (OPD) or ray

error, in each intermediate space, it is not clear which image point we are referring the

errors to. In the calculation of the OPD, in each intermediate space, which Gaussian

image point should be used to center our reference sphere on? And since generally we do

not have one unique exit pupil in the final image space if the system stop is not located in

this space, which coordinates are we using while we write out the wave aberration

function ? These difficulties might explain why more than 150 years have passed

since Seidel’s first description of his five Seidel aberrations, yet nobody has provided a

similar set of complete primary aberration coefficients for general anamorphic systems,

except in the simplest case [13] of a parallel-cylindrical anamorphic attachment system.

( , )W x y

1.3 Historical background

Part of the first-order theory of anamorphic systems was developed by Ernst Abbe at

the end of nineteenth century [19]. The research was continued by George J. Burch

(1904) [11], H. Chretien (1929) [12], C. G. Wynne (1954) [13], H. Kohler (1956) [15], K.

Bruder (1960) [16] and G. G. Slyusarev (1984) [18], who studied the aberrations of this

kind of system. Other authors have studied the general aberration features of anamorphic

Page 18: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

17

systems using the symmetry theory. For example, J. C. Burfoot (1954) [14], R. Barakat

and A. Houston (1966) [20], H. A. Buchdahl (1970) [1], P. J. Sands (1972) [3], and T. A.

Kuz'mina (1974) [21] considered the general aberration theory of double plane

symmetrical systems.

Among all the works on anamorphic system research mentioned above, special

attention is paid to C. G. Wynne’s paper "The Primary Aberrations of Anamorphic Lens

Systems" [13], published in 1954. In this paper, Wynne applied a modified traditional

trigonometric calculation method onto one of the simplest anamorphic systems, an

anamorphic attachment system composed of parallel-cylindrical lenses (with further

restriction on location of the system stop in the object space or final image space), and he

obtained the 16 primary aberration coefficient expressions for this attachment system, in

a form similar to the Seidel aberrations in RSOS. Other than this simplest case, K. Bruder

and G. G. Slyusarev [16,18] applied an angular eikonal method onto toroidal anamorphic

systems, but their development lack adequate theoretical structure, and accordingly the

results obtained are incomplete. Prior to the current work, the anamorphic primary

aberration theory remains a challenge for geometrical optics research.

From an optical design perspective, the primary aberration coefficient expressions

are extremely important in understanding the corresponding optical system because they

will help answer the following important questions:

1) What are the major errors in the system?

2) More importantly, what are the functional dependences of these errors so that we can

correct them?

Page 19: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

18

Additionally, we want all parameters in the primary aberration coefficient

expressions are first-order quantities so that the coefficients can be easily calculated. In

other words, we want the expressions are in a form similar to the Seidel aberrations.

The fact that we do not have a complete primary aberration theory for anamorphic

systems greatly limited our abilities to research the capacity of anamorphic image

formation. It also prevented us from obtaining insights in anamorphic imaging system

design research. The current work is dedicated to this important research area and will

address many questions arising from the above issues.

1.4 Dissertation content

This work will derive the complete monochromatic primary aberration coefficient

expressions for most anamorphic system types with double plane symmetry.

In chapter 2, we will explain the first-order theory needed for the development–these

will be the building blocks for the whole work.

Chapter 3 will be devoted to explicating the general aberration theory and the

aberration function model. In addition, it will identify which primary aberrations are

allowed for anamorphic systems.

In chapter 4, a general method used to find the primary aberration coefficients for

anamorphic systems will be built up.

Chapters 5 and 6 will be focusing on the most commonly used anamorphic systems:

systems made from cylindrical lenses. In chapter 5, the primary aberration coefficient

expressions for parallel cylindrical anamorphic attachment systems will be presented, and

Page 20: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

19

in chapter 6, the primary aberration coefficient expressions for cross cylindrical

anamorphic systems will be presented.

In chapter 7, the primary aberration coefficient expressions for toroidal anamorphic

systems will be presented. The primary aberration coefficient expressions for anamorphic

systems made from general double curvature surfaces, including fourth-order aspheric

departures, will be discussed in chapter 8.

In chapter 9, we will provide a testing scheme for the results obtained from chapters

5 through 8. In chapter 10, several design examples will be provided.

The conclusions and suggestions for future work will be given in chapter 11.

Page 21: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

20

CHAPTER 2

FIRST-ORDER THEORY FOR ANAMORPHIC SYSTEMS

The first-order theory is fundamental to understanding anamorphic systems. This

chapter will present the direction cosines of a ray and their paraxial approximations in

section 2.1; the three dimensional ray transfer and refraction equations in section 2.2; the

general theory of double curvature surfaces in section 2.3; the paraxial approximations of

double curvature surfaces in section 2.4; the ideal (first-order) image model for

anamorphic imaging in section 2.5; the paraxial ray tracing equations for anamorphic

systems in section 2.6; and the paraxial imaging properties of anamorphic systems in

sections 2.7 through 2.11.

2.1 Direction cosines of a ray and their paraxial approximations

The most convenient way to specify a ray, which is a straight line in homogenous

media, will be its direction cosines. Suppose that we have a system of Cartesian

coordinates with origin O as shown in Figure 2-1 below. Through the point O, we can

draw a line parallel to the ray whose direction is to be specified. Let us choose an

arbitrary point P on this parallel line through O, and project line OP onto the x, y, and z

axes, at points A, B, and C respectively. Then the three direction cosines components

of the ray to be specified are defined by ( , , )L M N

cos( ) ; cos( ) ; cos( )OA OB OCL AOP M BOP N COPOP OP OP

= ∠ = = ∠ = = ∠ = . (2-1)

Page 22: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

21

Figure 2-1 Direction cosines of a ray

Notice that of the three direction cosines components ( , of any ray, there are

only two components which are independent of each other, since we have the relationship

, )L M N

2 2 2 1L M N+ + = . (2-2)

From equation (2-2), we have

. (2-3) 2 2 1[1 ( )]N L M= − + / 2

Suppose axis z is the optical axis, and also suppose the ray to be specified lies in the

paraxial region close to the z axis. The domain of paraxial optics or first-order optics is

defined to be close enough to the optical axis to ensure that the ray angle and height

( ) are small quantities by first-order standards, whose squares and cross-

products are negligibly small for all surfaces

, , ,L M x y

j of the anamorphic optical system [22].

In the paraxial region, since and L M are small, we can expand equation (2-3) as a

binomial series

2 2( )1

2L MN + ...= − + . (2-4)

Page 23: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

22

To the first-order approximation, the quadratic terms in equation (2-4) can be ignored,

which shows andOC . 1N = OP=

Thus, in the paraxial region, equation (2-1) becomes

' 'cos( ) tan( ') ,

" "cos( ) tan( ") ,

1,

x

y

CP CPL AOP COP uOP OCCP CPM BOP COP uOP OC

N

⎧ = ∠ = = = =⎪⎪⎪ = ∠ = = = =⎨⎪

=⎪⎪⎩

(2-5)

here ,x yu u are tangent of the angles of the projections , made with the optical

axis z, respectively.

'OP ''OP

From the discussion above, we can draw the following conclusion: In the paraxial

region, the direction cosines of any ray are equal to the tangent of the angles formed by

the z axis and the projections of the ray in the respective x-z and y-z sections.

2.2 Three dimensional ray transfer and refraction equations

In optics, to trace an arbitrary ray through a system, we need two basic three

dimensional equations: The transfer equation and the refraction equation.

The transfer equation for a ray from a surface 1j − to the following surface j is

1 1 1

1 1 1

1j j j j j j

j j j

jx x y y z z tL M N

− − −

− − −

−− − − += = , (2-6)

here ( 1 1, , 1j j jx y z− − − ), ( , , )j j jx y z are the points where the ray intersects the previous

surface 1j − and the next surface j , respectively. 1jt − is the on-axis distance between the

Page 24: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

23

two surface vertices, and ( 1 1, , 1j j jL M N− − − ) are the ray direction cosines between the two

surfaces, as shown in Figure 2-2 below.

Figure 2-2 Three dimensional ray tracing

The three dimensional ray refraction equations are derived from Snell’s law [2].

Snell’s law states that the incident and refracted rays are coplanar with the surface normal

at the point of incidence, and they are related by

' sin ' sinn I n I= , (2-7)

here are the refractive indexes of the material before and after the refraction surface,

and

, 'n n

, 'I I are the angles that the incident and the refracted rays make with the surface

normal.

Snell’s law can be written in vector form as

'( ' ) ( )n n× = ×r n r n , (2-8)

here are unit vectors along the incident and refracted rays and n is the unit vector

along the normal of the surface at the point of incidence. By vectorially multiply on

both sides of Snell’s law, we get

r,r'

n

'[ ( ' )] [ ( )]n n× × = × ×n r n n r n . (2-9)

Page 25: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

24

Using the vector identity ( ) ( ) ( )× × = ⋅ − ⋅a b c b a c c a b we can rewrite equation (2-9)

as

'[ ' ( ' )] [ ( )]n n− ⋅ = − ⋅r n r n r n r n , (2-10)

this can be expanded in scalar form by setting ( , ,L M N ), ( ) and (', ', 'L M N , ,α β γ ) as

the components of and respectively, so that these quantities are direction cosines,

giving

, 'r r n

' ' ,' '' ' ,

n L nL kn M nM kn N nN k

α,β

γ

− =⎧⎪ − =⎨⎪ − =⎩

(2-11a)

where . '( ' ) ( ) 'cos ' cosk n n n I n I= ⋅ − ⋅ = −r n r n

It is often convenient to introduce the refraction operator , which signifies

refraction of the quantity operated upon, i.e.

Δ

( ) ' 'x xnu n u nuxΔ = − . If the quantity that

follows is a constant on refraction, then we will get zero. Taking the Lagrange

invariant as a common example, we will have

Δ

Ψ ( ) ' 0Δ Ψ = Ψ −Ψ = because Ψ is a

constant on refraction.

By using the refraction operator, equation (2-11a) can be rewritten as

( ) ,( )( ) .

nL knM knN k

,αβγ

Δ =⎧⎪Δ =⎨⎪Δ =⎩

(2-11b)

Suppose now the ray is refracted by surface j . Using equations (2-11a) and (2-11b),

and also noticing that 1' j jn n− = , 1' j jI I− = and ( 1 1' , ' , ' 1j jL M N j− − − ) = ( , ,j jL M N j ), we can

write the ray refraction equation on surface j as

Page 26: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

25

1 1 1 1 1 1

1 1cos cos

j j j j j j j j j j j jj

j j j

j j j j j

n L n L n M n M n N n Nk

k n I n I

α β γ− − − − − −

− −

− − −= = =

= −, (2-12)

here ( , ,j j jα β γ ) are the direction cosines of the thj surface normal. This is the ray

refraction formula we are going to use throughout this work, together with the ray

transfer equation (2-6).

2.3 Double curvature surface types and their surface normal—general theory

To form an anamorphic image, we need refractive (or reflective) surfaces with

double curvature inside our imaging systems. Additionally we need the double curvature

surfaces to be so aligned that the symmetry planes of the surfaces coincide with x-z and

y-z planes. This ensures our optical system will possess double plane symmetry, and we

can thus achieve different magnifications in the x-z and y-z symmetry planes. The optical

axis will be chosen as the line of intersection of the symmetry planes.

Now let us list several examples of existing double curvature surface types [23] so

that we can examine the basic concept of this kind of non-rotationally-symmetrical

surfaces.

From a mathematical point of view, the simplest double curvature surface type might

be an elliptical paraboloid with surface sag z expressed in Cartesian coordinates as

2 21 (

2)

x y

x yzr r

= + , (2-13)

Page 27: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

26

where xr and yr are the principal radii of curvatures in the x-z and y-z symmetry planes,

respectively. Cross sections of this surface parallel to the x-y plane are ellipses, and cross

sections perpendicular to that plane are parabolic.

From a fabrication point of view, the simplest double curvature surface type might be

a toroidal surface, described as

1 1

2 2 2 22[( ( ) ) ]x x y yz r r r r y x= − − + − − 2 , (2-14)

Again, xr and yr are the principal radii of curvature of the symmetry planes. The sag of

the toroidal surface to the fourth-order approximation is

2 2 4 2 2 4

3 21 1 2( ) (2 8x y x x y y

3 )x y x x y yzr r r r r r

= + + + + . (2-15a)

Notice that the most widely used double curvature surface type in current anamorphic

system design is the cylindrical surface, which is a special case of toroidal surfaces,

where one principal radius of curvature is equal to infinity. The surface sag equation for

cylindrical surfaces up to the fourth-order approximation is

2 4

3

1 12 8x x

x xzr r

= + . (2-15b)

Or,

2 4

31 12 8y y

yzr r

= +y . (2-15c)

Depending on which principal radius equals infinity.

From an optical testing perspective, the simplest double curvature surface type might

be the ellipsoid, as described by

Page 28: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

27

2 2 2

2 2 1x x y y

x y zr r r r

+ + = . (2-16)

The sag equation of this surface described for an oblate section and to the fourth-order

approximation is

2 2 4 2 2 4

3 2 21 1 2( ) (2 8x y x x y x y

)x y x x y yzr r r r r r r

= + + + + . (2-17)

From the above examples of existing double curvature surface types, we can see that

for this kids of surface, the surface sag equation can generally be written up to the fourth-

order approximation in a form

2 2 4 2 2 4

3 33 4 5

1 1 2( ) (2 8x y

3 )x y x x yzr r r r r

= + + + +y , (2-18)

here ,x yr r are the radii of curvature of the surface in the two principal sections, and

are certain coefficients with dimension of length. Notice that in this equation, we

permit the fourth-order aspheric departure in both principal sections by allowing

being different from

3 4 5, ,r r r

3 4 5, ,r r r

,x yr r .

We should notice that when the principal sections are in the form of circles (e.g. a

toroidal surface), we have 3 xr r= , 5 yr r= and . When , we

return to the special case of a spherical surface. Thus from equation (2-18), we can

achieve any double curvature surface types’ sag equation, up to the fourth-order

approximation, by properly choosing and .

2 1/ 34 ( )x yr r r= 3 4x yr r r r r= = = = 5

3 4, , ,x yr r r r 5r

Page 29: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

28

Now with the general double curvature surface sag-equation complete, it is necessary

to have convenient expressions for the direction cosines of the surface normal at the point

of incidence in order that the general ray refraction equation (2-12) can be applied.

Let us write the surface equation as ( , , ) 0F x y z = , then for a neighboring point

( , ,x x y y z zδ δ+ + +δ

0

) which is also on this surface [2], we will have

( , , )F x x y y z zδ δ δ+ + + = . (2-19)

By Taylor’s expansion, we can rewrite equation (2-19) as

( , , ) ... 0

0.

F F FF x y z x y zx y z

F F Fx y zx y z

δ δ δ

δ δ δ

∂ ∂ ∂+ + + +∂ ∂ ∂

∂ ∂ ∂⇒ + +

∂ ∂ ∂

= (2-20)

From this equation we see that the vector ( , ,F F F )x y z

∂ ∂ ∂∂ ∂ ∂

is perpendicular to the vector

( , , )x y zδ δ δ , and since the latter is restricted to lie in the surface, the

vector ( , ,F F F )x y z

∂ ∂ ∂∂ ∂ ∂

must be a vector along the surface normal at point ( , ,x y z ).

Therefore, the direction cosines of the surface normal are

12 2 2 2

( , , )( , , )

{( ) ( ) ( ) }

F F Fx y z

F F Fx y z

α β γ

∂ ∂ ∂∂ ∂ ∂=

∂ ∂ ∂+ +

∂ ∂ ∂

. (2-21)

2.4 Double curvature surfaces and their surface normal—paraxial approximations

Let us now consider the paraxial approximation of the general double curvature

surface equation (2-18) and the direction cosines of its surface normal.

Page 30: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

29

By ignoring terms higher than order three, we can rewrite the general double

curvature surface sag equation (2-18) as

2 21 (

2)

x y

x yzr r

= + . (2-22)

This equation shows that z is a quantity of the second order in ,x y which can be

neglected under the first-order approximation, so in the paraxial region, we have

0z = . (2-23)

This means in the paraxial region, surface sag can be ignored.

However, we need to remember that even though the surface sag can be taken as zero

in the paraxial domain, the surface still has curvature so that it has power to refract the

rays, thus we need to use equation (2-22) in the direction cosines of surface normal

calculation.

To find the direction cosines of surface normal in the paraxial region, we can rewrite

equation (2-22) as

2 21( , , ) ( ) 0

2 x y

x yF x y z zr r

= − + ≡ . (2-24)

By putting the second-order surface equation (2-24) into equation (2-21) and

ignoring any terms higher than order one in x and , we find the normal of the general

double curvature surface in the paraxial region as

y

(2-25) ,,

1,

x

y

c xc y

αβ

γ

⎧ = −⎪ = −⎨⎪ =⎩

Page 31: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

30

here 1/x xc r= , 1/y yc = r are the curvatures in both principal sections.

2.5 Ideal (first-order) image model for anamorphic imaging

Now let us see what kind of image the anamorphic system can provide. To do so we

will define an image model for the ideal behavior of anamorphic systems. This model

will be important because it will simplify the description of such systems by establishing

a reference. The model should be in accord with how an anamorphic system is ideally

meant to perform, and should be simple enough to provide insight.

A geometrical model for optical imagery that meets the above criteria would account

for the main features of the image. The fine details that are the departures from ideal

behavior can be described simply by a function depending on the aperture (stop) and field

(object) variables. This function, called the aberration function, is represented as a Taylor

series, and each term in the series represents a particular type of departure from ideal

behavior called an aberration. We will describe this function in Chapter 3.

To build our ideal imaging model, we will follow Abbe’s collinear mapping [22,24]

between two spaces: The object and the image space. The collinear mapping has the

following properties

1) Every object point will be mapped to a unique image point;

2) Every object plane will be mapped to a unique image plane.

From 1) and 2), we come to the conclusion that every object line will be mapped into a

unique image line. This result follows from the fact that a straight line is generated by the

intersection of two planes.

Page 32: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

31

Since imaging systems almost perform these functions, we will assume that their

behavior, including that of anamorphic systems, can be described by a collinear mapping.

The expressions that relate a point P ( , ,x y z ) in the object space to a point

('P ', ', 'x y z ) in the image space on a collinear mapping are

0000

1111'dzcybxadzcybxax

++++++

= , (2-26a)

0000

2222'dzcybxadzcybxay

++++++

= , (2-26b)

0000

3333'dzcybxadzcybxaz

++++++

= . (2-26c)

The spatial variables , ,x y z and ', ', 'x y z are positions in a Cartesian coordinate system

and the parameters are constants. Primed quantities refer to image space

quantities.

, , ,a b c d

Since an anamorphic system has double plane symmetry, we choose the z and 'z

axes to lie along the line of intersection of the symmetry planes and to act as our optical

axis, thus every surface will be centered about it. We also let the 'x axis be parallel to the

x axis and the axis to be parallel to the axis. In this manner, we can compose our

coordinate systems in both spaces. The only thing that has not been decided yet is the

location of the coordinate origins, and this will be discussed in Chapter 3.

'y y

We can now use the double plane symmetry condition to simplify the above collinear

mapping equations. Considering equation (2-26a), from the double plane symmetry

requirement, 'x remains unchanged when x stays the same but y changes in sign. This is

Page 33: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

32

true for all values of x and y, as long as 0 1 0b b= = . Also 'x must change sign when x

changes sign because of the double plane symmetry, hence 0 1 1 0a c d= = = . For equation

(2-26b), similarly we can show that of necessity 0 0 2 2 2 0a b a c d= = = = = . From equation

(2-26c), we notice that because of symmetry 'z does not change when x and y change

sign ( 'z is independent of x and y), and this requires that 0 0 3 3 0a b a b= = = = . Taking all

these requirements into account, equations (2-26) reduce to

1

0 0

' a xxc z d

=+

, (2-27a)

2

0 0

' b yyc z d

=+

, (2-27b)

3

0 0

' c z dzc z d

3+=

+. (2-27c)

By factoring out the constant and redefining the constants, we get 0d

1

0

2

0

3 3

0

' ,1

'1

' .1

a xxc z

b yyc zc z dzc z

⎧=⎪ +⎪

⎪=⎨ +⎪

⎪ +=⎪

,

+⎩

(2-28)

Equation (2-28) is the ideal (first-order) image model for anamorphic systems. From this

model, we know unless = , the imaging is anamorphic. The magnification in the x-

direction is given by

1a 2b

1

0 1xam

c z=

+, (2-29a)

Page 34: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

33

and the magnification in the y-direction is given by

2

0 1ybm

c z=

+. (2-29b)

The ratio of the two magnifications is often called the anamorphic ratio of the system.

So we have shown that anamorphic systems can indeed form anamorphic images

while satisfying the collinear mapping condition. We should remember that in the above

simplification, we did not put any restriction on where to locate the coordinate origins in

both spaces, and the origins in both spaces are not necessarily conjugate to each other.

The general feature of the ideal image model developed in this section is to allow

point to point mapping yet with an anamorphic image.

2.6 The paraxial ray tracing equations for anamorphic systems

From the discussions in the above sections, we know the passage of a ray through an

anamorphic system is governed by the refraction equation (2-12), the transfer equation

(2-6) and the surface equation (2-18). For any surface j in the anamorphic system under

consideration, in summary, we have

1 1 1

1 1 1

j j j j j j j

j j j

1x x y y z z tL M N

− − −

− − −

−− − − += = , (2-30a)

1 1 1 1 1 1j j j j j j j j j j j j

j j

n L n L n M n M n N n Nα β γ

− − − − − −− − −= =

j

, (2-30b)

2 2 4 2 2 4

3 3, , 3, 4, 5,

21 1( ) (2 8

j j j j j jj

x j y j j j j

x y x x y yz

r r r r r= + + + + 3 ). (2-30c)

From equations (2-5), (2-23) and (2-25), we know in the paraxial region,

Page 35: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

34

, ,

, ,

1

( , , ) ( , ,1),

( , , ) ( , ,1),

0.

j j j x j y j

j j j x j j y j j

j j

L M N u u

c x c y

z zα β γ

⎧ =⎪

= − −⎨⎪ = =⎩

Thus in this region, equations (2-30a)-(2-30b) can be rewritten as

1 11

, 1 , 1

j j j jj

x j y j

x x y yt

u u− −

−− −

− −= = , (2-31a)

, 1 , 1 , 1 , 11

, ,

(j x j j x j j y j j y jj j

x j j y j j

n u n u n u n un n

c x c y− − − −

− −= = − )− . (2-31b)

Equations (2-31) tell us an important fact— in the paraxial region surrounding the

axis of an anamorphic system, the ( , xx u ) and ( , yy u ) components of any paraxial ray

traced through the system are independent of one another, and each component can be

considered as if it is an independent paraxial ray traced in the x-z symmetry plane or y-z

symmetry plane of the system alone.

The conclusion is that in the paraxial region, rays can be traced by projecting into

two planes of symmetry, while the path of the project is governed entirely by the normal

law of paraxial ray tracing and the paraxial curvature ,x yc c in both symmetry planes.

To clearly emphasize this, we can write equations (2-31a)-(2-31b) into the

independent ray trace equations separately. For ( , xx u ) components of this paraxial ray,

we have

1 1 , 1j j j x jx x t u− − −− = , (2-32a)

, 1 , 1 1( ) ,j x j j x j j j j x jn u n u x n n c− − −− = − − . (2-32b)

Page 36: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

35

Notice that these two equations are in exactly the same form of the paraxial meridian ray

tracing equations for a RSOS made from spherical surfaces 2 2, ( ) / 2 ...j x j j jz c x y= + + , in

which the x-z symmetry plane will be a meridian section. So we can imagine that we

have a RSOS associated with the x-z symmetry plane of the anamorphic system, and we

call it the associated x-RSOS.

For ( , yy u ) components of this paraxial ray, we have

1 1 , 1j j j y jy y t u− − −− = , (2-33a)

, 1 , 1 1( ) ,j y j j y j j j j y jn u n u y n n c− − −− = − − . (2-33b)

We see that these two equations are in exactly the same form of the paraxial meridian ray

tracing equations for another RSOS made from spherical surfaces

, in which the y-z symmetry plane will be a meridian section. So we can imagine that

we have another RSOS associated with the y-z symmetry plane of the anamorphic system,

and we can call it the associated y-RSOS.

2 2, ( )j y j j jz c x y= + / 2

...+

Equations (2-32) and (2-33) are the basic paraxial ray tracing equations for an

anamorphic system.

2.7 Paraxial image properties of anamorphic systems---part one

From the discussion in section 2.6, since for any arbitrary paraxial ray (either skewed

or non-skewed) in an anamorphic system, its ( ,,j x jx u ) component and ( ,,j y jy u )

component are completely independent of each other and are traced through the system

by projecting into the x-z and y-z symmetry planes according to their own ray trace

Page 37: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

36

equations (2-32) and (2-33) , we arrive at a very important conclusion: whenever we are

working with an anamorphic paraxial ray’s ( , xx u ) component, we can imagine that we

are dealing with the projection of this paraxial ray in the x-z symmetry plane. This

projection can be further imagined as a paraxial ray, staying in the x-z meridian plane of

the associated x-RSOS. Thus, all results from Gaussian optics for the associated x-RSOS,

can be applied to the ( , xx u ) component of this anamorphic paraxial ray directly except

that every quantity will now have a subscript x , including the x-paraxial object plane

location xl , the x-paraxial entrance pupil location xl , the x-paraxial marginal ray angle xu

and height xh , the x-paraxial chief ray angle xu and height xh , etc [2]. Figure 2-3 shows

these quantities in an intermediate space.

Figure 2-3 Gaussian optics properties for the associated x-RSOS (in an intermediate space)

Similarly, whenever we are dealing with the ( , yy u ) component of the same

anamorphic paraxial ray, we can imagine that we are dealing with the projection of this

Page 38: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

37

paraxial ray in the y-z symmetry plane of our anamorphic system. Again, this projection

can be further imagined as a paraxial ray, staying in the y-z meridian plane of the

associated y-RSOS. Thus all results from Gaussian optics of the associated y-RSOS can

be applied to ( , yy u ) component of this anamorphic paraxial ray directly except every

quantity will now have a lower subscript . y

2.8 Paraxial image properties of anamorphic systems---part two

It is well know that in RSOS there are only two independent paraxial rays. Normally

they are taken to be the marginal ray and chief ray, and any third paraxial ray can be

written as a linear combination of these two [22,25].

Similarly, in an anamorphic system, from equations (2-32) and (2-33), we can prove

there are only two linearly independent paraxial skew rays too.

To prove this, suppose we have two known paraxial skew rays with components on

surface j as ( 1, 1 ,,j x jx u ), ( 1, 1 ,,j y jy u ) and ( 2, 2 ,,j x jx u ), ( 2, 2 ,,j y jy u ). The paths of these two

skew rays through the system have been completely determined by equations (2-32) and

(2-33). Suppose that we also have a third unknown paraxial ray, for which we denote the

relevant components as ,( , )j x jx u , ,( , )j y jy u on surface j .

Suppose we can write the third unknown paraxial ray’s ,( , )j x jx u component as

combinations of the two known paraxial rays’ ( , xx u ) components in the form

, 1, , 2,j x j j x j jx C x D x= + , (2-34a)

, , 1 , , 2 ,x j x j x j x j xu C u D u j= + , (2-34b)

Page 39: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

38

here , , ,x j xC D j are proportional constants on surface j and we can obtain the values of

them by solving these equations.

If we can prove that , , ,x j xC D j have no dependence on surface number j and are

constants throughout the anamorphic system, then we know for this third unknown

paraxial ray, its ( , )xx u component can always be expressed as a linear combination of

the two known paraxial skew rays. And if the same thing is true for its ( , )yy u component,

we immediately know the third ray can not be independent to the two known paraxial

rays from linear algebra theory. By this way, we can prove that there are only two

linearly independent paraxial skew rays in any anamorphic system.

To prove it, let us rewrite paraxial ray trace equations (2-32a), (2-32b) into the

following form

1 , 1 1j j x j jx x u t− − −= + , (2-35a)

, 1 , 1 1( ) ,j x j j x j j j j x jn u n u x n n c− − −= − − . (2-35b)

Equations (2-35) hold for ,( , )j x jx u also because the third ray is also a paraxial ray in our

anamorphic system, thus by substituting equations (2-34) into equations (2-35), we have

1 , , 1, , 2, , 1 , , 2 ,

, 1, 1 , , 2, 2 ,

, 1, 1 , 2, 1

( ) (

( ) ( )

,

j j x j j x j j x j j x j x j x j x j j

x j j x j j x j j x j j

x j j x j j

)x x u t C x D x C u D u t

C x u t D x u t

C x D x

+

+ +

= + = + + +

= + + +

= +

1 , 1 , 1 1 , 1

, 1 , , 2 , , 1, 1 , 2, 1 1 , 1

, 1 , 1, 1 1 , 1 , 2 , 2, 1 1 ,

, 1 1 , 1 , 1 2

( )( ) ( )( )

[ ( ) ] [ ( ) ]

j x j j x j j j j x j

j x j x j x j x j x j j x j j j j x j

x j j x j j j j x j x j j x j j j j x j

x j j x j x j j x

n u n u x n n cn C u D u C x D x n n cC n u x n n c D n u x n n cC n u D n u

+ + + + +

+ + + +

1+ + + + +

+ + +

= − −

= + − + −

= − − + − −

= + , 1j+

+

Page 40: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

39

1 , 1 , 1 , 2 , 1( )j x j x j x j x jn C u D u+ += + .+

Thus we find

1 , 1, 1 , 2, 1j x j j x j jx C x D x+ + += + , (2-36a)

, 1 , 1 , 1 , 2 , 1x j x j x j x j xu C u D u j+ + += + . (2-36b)

Notice that the same process can be continued to the next surface and so on.

By comparing equations (2-36) with equations (2-34), we see the proportionality

constants , , ,x j xC D j do not change with the passage of the third paraxial ray through the

anamorphic system. Hence they are indeed constant throughout the anamorphic system

and we will denote them as ,x xC D because their values do not depend on surface

number j .

Thus, we have proved that the third ray’s ,( , )j x jx u component can be written as a

liner combination of the two known paraxial skew rays’ ( 1, 1 ,,j x jx u ), ( 2, 2 ,,j x jx u )

components for any surface number j in the anamorphic system.

Exactly the same way, we can prove that we can always write the unknown third

paraxial ray’s ( ,j jy u ) component as a linearly combination of the two known paraxial

skew rays’ ( 1, 1 ,,j y jy u ), ( 2, 2 ,,j y jy u ) components on surface j , as

1, 2,j y j y jy C y D y= + , (2-36c)

, 1 , 2 ,y j y y j y yu C u D u j= + . (2-36d)

Again, here ,y yC D are proportional constants and they are constants throughout the

system.

Page 41: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

40

These four equations in (2-36) show there are only two linearly independent paraxial

skew rays in an anamorphic system, all other paraxial rays can be expressed as linear

combinations of these two known paraxial rays, with different proportionality

constants , , ,x x y yC D C D .

In practice, the two known paraxial skew rays are often taken as a paraxial skew

marginal ray (which comes from the on-axis object point and passes through a point on

the edge of the system stop) and a paraxial skew chief ray (which comes from a point on

the maximum object field and passes through the center of the stop). Once we know these

two rays, we can use their different linear combinations to form all other paraxial rays in

the anamorphic system.

2.9 Paraxial image properties of anamorphic systems---part three

In section 2.8, we obtained the linear combination relationships between any third

paraxial rays with the two known paraxial skew rays in an anamorphic system. There also

exists particular relationships between the two known paraxial rays—the anamorphic

Lagrange invariants [1], similar to the Lagrange invariant relationship in RSOS [2,22].

For the two known linearly independent paraxial skew rays’ ( ,,j x jx u ) components,

from equations (2-35), we have

1, 1, 1 1 , 1 1j j x j jx x u t− − −= + , (2-37a)

1 , 1 1 , 1 1, 1 ,( )j x j j x j j j j x jn u n u x n n c− − −= − − , (2-37b)

2, 2, 1 2 , 1 1j j x j jx x u t− − −= + , (2-37c)

Page 42: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

41

2 , 1 2 , 1 2, 1 ,( )j x j j x j j j j x jn u n u x n n c− − −= − − . (2-37d)

From equations (2-37), we have

2 , 1, 1 , 2, 2 , 1, 1 , 2, 1, 2, 1 , 1, 2, 1 ,

2 , 2, 1 , 1, 1 , 1, 1 , 2,

1 2 , 1 1, 1 1 , 1 2,

1 2 , 1

( ) ( )

[ ( ) ] [ ( ) ]

(

j x j j j x j j j x j j j x j j j j j j x j j j j j x j

j x j j j j x j j j x j j j j x j j

j x j j j x j j

j x j

n u x n u x n u x n u x x x n n c x x n n c

n u x n n c x n u x n n c xn u x n u x

n u

− −

− −

− − − −

− −

− = − + − − −

= + − − + −

= −

= 1, 1 1 , 1 1 1 1 , 1 2, 1 2 , 1 1

1 2 , 1 1, 1 1 1 , 1 2, 1

) ( )

...tan .

j x j j j x j j x j j

j x j j j x j j

x

x u t n u x u t

n u x n u x

cons t λ

− − − − − − − −

− − − − − −

+ − +

= −

== =

Hence for all surfaces, we have

1 2 2 1( ) tanx xn x u x u cons t xλ− = = . (2-38a)

Equation (2-38a) shows the connection between the projections of the two known

paraxial skew rays in x-z symmetry plane of the anamorphic system, and it is very similar

to the Lagrange invariant relationship in the associated x-RSOS.

Exactly the same way, we can find

2 , 1, 1 , 2, 1 2 , 1 1, 1 1 1 , 1 2, 1

...tan .

j y j j j y j j j y j j j y j j

y

n u y n u y n u y n u y

cons t λ

− − − − − −− = −

== =

Hence for all surfaces, we also have

1 2 2 1( ) tany yn y u y u cons t yλ− = = . (2-38b)

Equation (2-38b) shows the connection between the projections of the two known

paraxial skew rays in y-z symmetry plane, and it is very similar to the Lagrange invariant

relationship in the associated y-RSOS.

Page 43: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

42

When the constants ,x yλ λ are at their maximum possible value, we will replace them

as ,x yΨ Ψ (the Lagrange invariants associated with the two associated RSOS), called the

x-Lagrange invariant and the y-Lagrange invariant, respectively, which differ from ,x yλ λ

by some proportional constants.

Before we go to the next step, it might be an appropriate time to summarize the

difference between the paraxial optics for RSOS and for anamorphic systems.

For RSOS, all possible paraxial marginal rays from an on-axis object point are the

same because of the rotational symmetry, thus we can take the paraxial marginal ray as

the one lying in the meridian plane (the plane containing the optical axis and the object

point). Similarly, we can take the paraxial chief ray as the one staying in the meridian

plane also, thus RSOS ray tracing can be reduced to ray tracing in the meridian plane.

But for anamorphic systems, in general, if a paraxial marginal ray does not stay in

one of the symmetry planes, it will be a skew ray whose passage through the system will

not be confined in any single plane. Similarly, a paraxial chief ray will generally be a

skew ray unless the object point stays in one of the symmetry planes. Because of these

complications, we cannot reduce the anamorphic paraxial ray tracing into a ray trace

within a single meridian plane. Instead, we need to trace a skew paraxial marginal ray

and a skew paraxial chief ray in order to fully specify the paraxial anamorphic system.

In practice, it is not so convenient to fully specify the paraxial anamorphic system

using two skew paraxial rays. Thus we need to go a step further.

From the discussion in section 2.6, we know for the skew paraxial marginal ray or

chief ray, the ray tracing can be done by projecting onto the x-z symmetry plane and the

Page 44: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

43

y-z symmetry plane of the anamorphic system. These projections can be considered as

independent paraxial rays traced in the associated x-RSOS and y-RSOS.

In each RSOS, we know there are only two independent paraxial rays [22,25],

normally they are taken to be the non-skew marginal ray and chief ray traced in the

meridian plane, and any third paraxial ray can be written as a linear combination of these

two.

Because each projection of the skew paraxial marginal ray or chief ray in the

corresponding symmetry plane can be expressed by the non-skew marginal ray and chief

ray in the corresponding associated RSOS, the skew paraxial ray can be fully specified by

four non-skew paraxial rays, namely the x-marginal ray, the x-chief ray, the y-marginal

ray, and the y-chief ray, in the corresponding x-RSOS and y-RSOS.

So we can trace the x-marginal ray and x-chief ray in the x-z symmetry plane and the

y-marginal ray and y-chief ray in the y-z symmetry plane, and then use all four non-skew

paraxial rays to fully specify the paraxial anamorphic system.

2.10 Paraxial image properties of anamorphic systems---part four

From the discussion in section 2.9, we know that even though there are only two

linearly independent paraxial skew rays in an anamorphic system and any other paraxial

ray can be written as a linear combination of these two rays, it is actually more

convenient to make use of four separately known non-skew paraxial rays—the paraxial

marginal and chief rays associated with the x-RSOS, traced in the x-z meridian plane;

Page 45: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

44

and the paraxial marginal and chief rays associated with the y-RSOS, traced in the y-z

meridian plane.

When we are dealing with an arbitrary anamorphic paraxial (either skewed or non-

skewed) ray’s ( , xx u ) components, we will make use of the x-marginal and x-chief rays

which lie in the x-z symmetry plane of our anamorphic system. Similarly, whenever we

are dealing with the ( , yy u ) component of the same anamorphic paraxial ray, we will

make use of the y-marginal and y-chief rays which stay in the y-z symmetry plane of the

anamorphic system.

Let us write , ,( , )x j x jh u , , ,( , )x j x jh u as the parameters associated with the x-marginal

and x-chief ray in x-RSOS, and also write , ,( , )y j y jh u , , ,( , )y j y jh u as the parameters

associated with the y-marginal and y-chief rays in y-RSOS, on surface j . From equations

(2-36) and (2-38), we have

,j x x j x x , jx C h D h= + , (2-39a)

, , ,x j x x j x xu C u D u j= + , (2-39b)

,j y y j y yy C h D h= + , j , (2-39c)

, , ,y j y y j y yu C u D u j= + , (2-39d)

, , , ,( )j x j x j x j x j xn h u h u− = Ψ , (2-39e)

, , , ,(j y j y j y j y j yn h u h u )− = Ψ . (2-39f)

, , ,x y xC C D Dy are proportionality constants throughout the system, and they can be found

by the arbitrary anamorphic paraxial ray’s initial condition. xΨ is the x-Lagrange

Page 46: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

45

invariant associated with the x-RSOS made from imaginary spherical

surfaces . 2 2, ( ) / 2 ...j x j j jz c x y= + + yΨ is the y-Lagrange invariant associated with the

corresponding y-RSOS made from imaginary spherical surfaces

.

2 2, ( )j y j j jz c x y= + / 2

...+

Now let us find the value of the proportionality constants , ,x yC C Dx and yD . Suppose

the arbitrary anamorphic paraxial ray cuts the paraxial object plane ( 0j = ) at point

( 0 0,ξ η ). Let be the distance from this object point to the optical axis,

and let the polar angle be

2 2 1/0 0(d ξ η= + 2)

φ . It then follows 0 0cos , sind dξ φ η φ= = , as shown in Figure

2-4 below. Suppose the maximum object field is , which is given by the radius of the

object point farthest from the origin in object coordinates, and we define the

quantity as the fractional field (object).

maxd

max/H d d=

Let this ray cut the system stop plane ( j p= ) at point ( ,p px y ), let

be the distance from this point to the optical axis, and let the polar angle be

2 2 1( )p pe x y= + / 2

θ . It then

follows that cos , sinp px e y eθ θ= = , as shown in Fig 2-4 below. Suppose the maximum

aperture is , and we define the quantitymaxe max/e eρ = as the fractional aperture (stop).

From equations (2-39), we know, at the paraxial object plane ( 0j = ), we have

0 ,0 ,0x x x xx C h D h= + .

In this plane, the x-marginal ray height ,0xh =0, the x-chief ray height ,0 maxxh d= and

0 0 cosx dξ φ= = . So we find that

Page 47: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

46

0 ,0 max/ cos / cosx x xD h d d Hξ φ φ= = = = H ,

here cosxH H φ= is the projection of the fractional field in the x-z symmetry plane of

the anamorphic system, for this arbitrary paraxial ray.

H

Figure 2-4 The object and stop plane

At the system’s stop plane ( j p= ), we will have

, ,p x x p x x px C h D h= + .

In this plane, the x-marginal ray height , mx ph e ax= , the x-chief ray height ,x ph =0 and

cosp px x e θ= = . So we find that

, max/ cos / cosx p x p xC x h e eθ ρ θ ρ= = = = ,

here cosxρ ρ= θ is the projection of the fractional aperture ρ in the x-z symmetry plane,

for this ray.

Thus, we have found the proportionality constants ,x x xC D xHρ= = , which are

constants throughout the system. For any surface j , equations (2-39a)-(2-39b) become

, ,j x x j x x jx h H hρ= + , (2-40a)

Page 48: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

47

, , ,x j x x j x xu u H u jρ= + . (2-40b)

Similarly, we can find the proportional constants ,y yC ρ= y yD H= . For any

surface j , equations (2-39c)-(2-39d) become

, ,j y y j y yy h H hρ= + j , (2-40c)

, , ,y j y y j y yu u H u jρ= + , (2-40d)

here sinyρ ρ= θ is the projection of the fractional aperture ρ in the y-z symmetry plane,

and sinyH H φ= is the projection of the fractional field in the y-z symmetry plane, for

this ray.

H

We recognize that ( ,x yH H ) and ( ,x yρ ρ ) are actually the normalized object and

aperture coordinates of this arbitrary anamorphic paraxial ray.

Equations (2-40) will serve as the foundation of primary aberration coefficients

derivation for anamorphic systems. These equations can be understood this way:

1) In an anamorphic system, the ray trace data of any anamorphic paraxial ray (either

skewed or non-skewed) can be composed by the linear combinations of the four known

non-skewed paraxial marginal and chief rays’ tracing data in the two associated RSOS.

2) Further more, the proportionality constants are the normalized object and stop

coordinates of this arbitrary anamorphic paraxial ray under study. Also notice that when

we are exploring the object and stop planes, these coordinates because variables.

Thus, all paraxial quantities in an anamorphic system can be written in terms of the

four known non-skewed paraxial rays’ tracing data in the two associated RSOS, together

with object and stop variables.

Page 49: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

48

2.11 Paraxial image properties of anamorphic systems---part five

In the above sections, we have already composed the essential equations of the

paraxial optics for anamorphic systems. In this section, we are going to build up the

necessary paraxial definitions which will be extensively used in primary aberration

calculations throughout this work.

From equation (2-32b) we can get the refraction invariant of the x-marginal ray, for

the associated x-RSOS. To show this, we rewrite (2-32b) for the marginal ray as

1 , 1 1 , , , , , ,j x j j x j x j j x j j x j x j x jn u n h c n u n h c A− − −+ = + = . (2-41a)

Thus, we see that the quantity ,x jA is an invariant on refraction for the x-marginal ray, in

the associated x-RSOS, on surface j . Similarly, the refraction invariant for the x-chief

ray on surface j is

1 , 1 1 , , , , , ,j x j j x j x j j x j j x j x j x jn u n h c n u n h c A− − −+ = + = . (2-41b)

Similarly, from the equation (2-33b), we know the refraction invariants for the

associated y-RSOS, on surface j , are

1 , 1 1 , , , , , ,j y j j y j y j j u j j y j y j y jn u n h c n u n h c A− − −+ = + = , (2-41c)

1 , 1 1 , , , , , ,j y j j y j y j j y j j y j y j y jn u n h c n u n h c A− − −+ = + = . (2-41d)

By applying equations (2-41), from equations (2-39e)-(2-39f), the Lagrange

invariants associated with the two RSOS become

, , , , , , , ,( )x j x j x j x j x j x j x j x j x jn h u h u A h A hΨ = − = − , (2-42a)

, , , , , , , ,( )y j y j y j y j y j y j y j y j y jn h u h u A h A hΨ = − = − . (2-42b)

Page 50: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

49

2.12 Summary

In section 2.1, we found that in the paraxial region, we have the direction

cosines . In section 2.2, we provided the three dimensional ray-

tracing equations. In section 2.3, we found that for any double curvature surface, its’ sag

equation can generally be written in a form as

, ,x yL u M u N= = 1=

2 2 4 2 2 4

3 3 33 4 5

1 1 2( ) ( ) ...2 8x y

x y x x y yzr r r r r

= + + + + + .

In section 2.4, we found that in the paraxial region, the direction cosines of the

surface normal for any double curvature surface are , ,x yc x c y 1α β γ= − = − =

) ;

. In section

2.5, we found from the collinear mapping point of view, anamorphic imaging with point-

to-point mapping is possible. In section 2.6, we found that in the paraxial region, the 3D

skew ray tracing equations for anamorphic systems can be replaced by the four 2D non-

skew ray-tracing equations in the two symmetry planes

1 1 , 1 , 1 , 1 1 ,, (j j j x j j x j j x j j j j x jx x t u n u n u x n n c− − − − − −− = − = − −

1 1 , 1 , 1 , 1 1 ,, ( )j j j y j j y j j y j j j j yy y t u n u n u y n n c− − − − − −− = − = − − j .

From section 2.7 to section 2.11, we found that we can think of a paraxial

anamorphic system as the two associated RSOS, thus all results we known for the two

RSOS can be applied to the anamorphic system directly. We also found that there are

only two independent paraxial skew rays in an anamorphic system, but we prefer using

the four non-skew marginal and chief rays traced in the associated x-RSOS and y-RSOS

to fully specify the system. We found that by using these four non-skew paraxial rays, we

can get all paraxial quantities associated with the anamorphic system.

Page 51: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

50

All results built up in this chapter will be integrated into the anamorphic primary

aberrations development in chapters 3 through 8.

Page 52: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

51

CHAPTER 3

GENERAL ABERRATION THEORY FOR ANAMORPHIC SYSTEMS

In chapter 2.5, we constructed the ideal (first-order) image model for an anamorphic

imaging system which accounts for the main features of the image. In this chapter, we

will build the aberration function, which is the departure from ideal behavior, and thus

accounts for the fine details. This departure can be described by a function depending on

the aperture (stop) and field (object) variables. We will write the aberration function into

its power series expansion form and show that there are sixteen primary aberration types

in an anamorphic system. We will also demonstrate the connection between anamorphic

primary wave error and ray errors.

We will present Fermat’s principle and Sir Hamilton’s characteristic function in

section 3.1, aberration function and ray aberrations for anamorphic systems in section 3.2,

and the power series expansion of aberration function in section 3.3.

3.1 Fermat’s principle and Sir Hamilton’s characteristic function

One of the basic laws of Geometrical optics is Fermat’s principle, which states: a

light ray traveling from point P to point P’ must traverse an optical path length that is

stationary with respect to variations of that path [26]. From Fermat’s principle, we can

draw an important conclusion: for any two non-conjugate points P and P’ in an optical

system, there will be one and only one ray that passes through both points. This

Page 53: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

52

conclusion is invalid if P and P’ are conjugate points since all rays passing through the

conjugate points will have the same optical path length.

The theoretical importance of this conclusion is that for any ray which comes from a

point in the object plane and passes through a point in the system stop plane, this ray will

be completely defined by these two points, thus every parameter of this unique ray can be

written as a function of the object and the stop coordinates, and can be further expressed

as a power series expansion using these four variables. We will use this concept soon in

this chapter.

Now let us assume P( , ,x y z ) and P’( ', ', 'x y z ) are any two non-conjugate points in

our anamorphic optical system. We know there will be a unique ray that passes through

both points. From Sir Hamilton’s characteristic function theory [1-2,27], we know the

Hamilton’s point characteristic function V is defined as the optical path length along this

unique ray from P to P’ and it can be written as

( , ') ( , , , ', ' ')V P P V x y z x y z= . (3-1)

This function is of great theoretic value since the direction cosines of this ray

are totally determined by it via the following relationships

( , ')V P P

' ''

Vn Lx∂

=∂

, (3-2a)

' ''

Vn My∂

=∂

, (3-2b)

' ''

Vn Nz∂

=∂

, (3-2c)

here ' is the associated refractive index in the space where point P’ is located. n

Page 54: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

53

3.2 Aberration function and ray aberrations for anamorphic systems

In chapter 2.5 we obtained the ideal image model for anamorphic systems. We will

now construct the aberration function to account for departures from the ideal behavior.

To do this, the symmetry of the system will be invoked to determine the possible

aberration form.

Remember in our ideal anamorphic imaging model, we did not put any restriction on

where to locate the coordinate origins in both object and image spaces. In RSOS, the

common practice is to locate the coordinate origins at the system’s entrance and exit

pupil, and the pupil coordinates are then used to define the system aberration function.

But in an anamorphic imaging system, as discussed in chapter 1, because the x-pupil and

y-pupil generally will not coincide with each other, we do not have such natural choices

to serve as our coordinate origins.

In this work, we will arbitrarily define the plane tangential to the last refraction

surface in the final image space as our image space reference plane, and it will play the

same role as the exit pupil plane played in RSOS. On this plane, we will build up our x-y

coordinates, which are centered on the system optical axis at point O. In object space, we

choose the reference plane as the object plane itself.

Using the above defined coordinate origins, consider the following anamorphic

imaging system: Suppose we have an object point , in the paraxial object plane.

Suppose point

0 0( , )P x y

0 0 0'( , )P ξ η is the ideal image point in the final image space. Let 'Σ be the

wavefront of the ray bundle from P which passes through the coordinate origin O, and

Page 55: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

54

let S be a reference sphere with center and radius . Let another ray r of the ray

bundle from meet and

0 'P 0 'P O

P S 'Σ at the points , respectively, and let it meet the final

image plane at point . The coordinates of will be ( ,

0,Q Q

'P 0, 'Q P , )x y z , ( ,ξ η ) respectively,

and the direction cosines of the ray r will be . ( , , )L M N pz is the distance from the last

surface to the final image plane, as shown in Figure 3-1 below.

Figure 3-1 Wavefront error

Notice that the skew paraxial chief ray (which pass through the center of system stop)

no longer passes through pointO since the x-y plane is arbitrarily chosen to lie on the

tangential to the final refraction surface, and since the chief ray will be skewed with

respect to the optical axis if it does not stay in one of the symmetry planes.

The wave aberration function, is defined as the optical path length from the

reference sphere to the wavefront

( , )W x y

S 'Σ , measured along the ray as a function of the

transverse coordinates ( ,x y ) of the ray intersection with a reference sphere centered on

the ideal image point [2].

Page 56: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

55

From Sir Hamilton’s characteristic function theory, we know

0 0 0

0

( , ; , ) ( , ) ( , )( , ) ( , ),

W x y x y V P Q V P QV P O V P Q

= −= −

(3-3a)

because Q and O are on the same wavefront and are therefore at the same optical distance

from P. Differentiating equation (3-3a) and noticing that the point characteristic function

is a constant with respect to reference sphere variables ( , )V P O x and , we get y

W V V zx x zW V V

xz

y y z y

∂ ∂ ∂ ∂= − − ⋅

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

= − − ⋅∂ ∂ ∂ ∂

(3-3b)

The equation of the reference sphere is

2 2 2 2 2

0 0 0 0

2 2 20 0

( ) ( ) ( )

2 2 2 0.p

p

2px y z z

x y z x y z z

ξ η ξ η

ξ η

− + − + − = + +

⇒ + + − − − =

z (3-3c)

From the reference sphere equation we can find

0

0

( ) ,

( ) .

p

p

z xx z zz yy z z

ξ

η

∂ −⎧ =⎪∂ −⎪⎨∂ −⎪ =⎪∂ −⎩

(3-3d)

By inserting equation (3-3d) into equation (3-3b) and taking equations (3-2) into account,

we get

0

0

1 '([ ' ],'

1 '([ ' ]'

p

p

W N xLn x z z

W N yMn y z z

ξ

η

∂ −⎧ = − +⎪ ∂ −⎪⎨ ∂ −⎪ = − +⎪ ∂ −⎩

)

) . (3-3e)

But from the definition of direction cosines, we know

Page 57: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

56

0

( , ,( ', ', ')

'p )x y z z

L M NQ P

ξ η− − −= . (3-3f)

This simplifies equation (3-3e) into

0 0

0 0 0 0

0 0

0 0 0 0

1 )( )' ' ' '

1 ( )' ' ' '

W x xn x Q P Q P Q P Q P

W y yn y Q P Q P Q P Q P

,'

,'

ξ ξ ξ ξ δξ

η η η η δη

∂ − − −⎧ = − + = − = −⎪ ∂⎪⎨ ∂ − − −⎪ = − + = − = −⎪ ∂⎩

(3-3g)

here (δξ ,δη ) are ray errors, by definition.

From above deduction, we get the relationship between wavefront error and ray

errors in reference sphere coordinates as

0

0

' ,'

' .'

Q P Wn x

Q P Wn y

δξ

δη

∂⎧ = −⎪ ∂⎪⎨ ∂⎪ = −

∂⎪⎩

(3-4a)

When we are dealing with primary (third-order) aberrations, we can replace the

unknown distance by0 'Q P R , the radius of the reference sphere. And we can replace

reference sphere coordinates by reference plane coordinates [2,28]. Thus equation (3-4a)

becomes

,

'

.'

R Wn xR Wn y

δξ

δη

∂⎧ = −⎪ ∂⎪⎨ ∂⎪ = −

∂⎪⎩

(3-4b)

Thus we see in anamorphic systems, the ray aberrations are proportional to the derivative

of wave aberration function , which is the same relationship as found in RSOS

with image space coordinate origin located on the system exit pupil plane.

( , )W x y

Page 58: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

57

Suppose the reference plane is surface k in the optical system above. We then can

express the aberration functionW in the coordinates of the reference plane in the image

space as . ( , )k kW x y

Suppose now the ray r as shown in Figure 3-1 meets the system stop at a point of

fractional aperture ( ,x yρ ρ ) and also suppose the fractional field is ( ,x yH H ). From the

conclusion of Fermat’s principal, as discussed in Section 3.1, we know all parameters

associated with this ray r can be expressed as a function of ( ,x yH H , ,x yρ ρ ) and can be

further written as the Taylor expansion of these four quantities. In other words, kx , ky

and the aberration function W can be written as a Taylor series expansion of

( ,x yH H , ,x yρ ρ ).

For the primary aberration calculation, we can evaluate kx and ky as their paraxial

equivalents [22,28], thus according to equations (2-40), we have

, ,

, ,

,

,k x x k x x k

k y y k y y k

x h H h

y h H h

ρ

ρ

⎧ = +⎪⎨

= +⎪⎩

here ( , ,, ,x k y kh h ), ( , ,,x k y kh h ) are paraxial marginal and chief ray data in the two

corresponding associated RSOS, as defined in chapter 2.10.

Notice that the parameters ,x yH H can be thought of as constants when we are

exploring the aperture (stop), thus we have

,

,

1 ,

1 .

x

k x k x k

y

k y k y k

W W Wx x h

W W Wy y h

x

y

ρρ ρ

ρρ ρ

∂ ∂ ∂ ∂= ⋅ = ⋅

∂ ∂ ∂ ∂

∂∂ ∂ ∂= ⋅ = ⋅

∂ ∂ ∂ ∂

Page 59: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

58

By putting these results into equation (3-4b), we get

, , ,

, , ,

1 1 ,' '( / ) ' '

1 1 .' '( / ) ' '

x k x x k x x k x

y k y y k y y k y

R W Wn h n h R n u

W

R W Wn h n h R n u

δξ

Wρ ρ ρ

δηρ ρ ρ

∂ ∂⎧ = − = − =⎪ ∂ ∂⎪⎨ ∂ ∂⎪ = − = − =⎪ ∂ ∂⎩

∂∂

∂∂

These equations are accurate for primary aberration calculation purpose. After regrouping,

we get the finally relationship between primary wavefront error and ray error, in stop

plane coordinates as

,

,

' ' ,

' ' .

x kx

y ky

Wn u

Wn u

δξρ

δηρ

∂⎧ =⎪ ∂⎪⎨ ∂⎪ =

∂⎪⎩

(3-4c)

Equation (3-4c) tell us that we can now calculate the anamorphic primary ray errors

using the stop coordinates, instead of the coordinates of the arbitrarily defined image

space reference plane.

3.3 Power series expansion of aberration function

From the discussion in section 3.2, all parameters with the ray path r will be a

function of the normalized object and stop variables ( ,x yH H , ,x yρ ρ ), and thus the wave

aberration function W in general will be a function of the four

variables ( , , , )x y xH H yρ ρ

)

also. As a result, we can either write the wave aberration

function as or ( , )k kW x y ( , , ,x y x yW H H ρ ρ . If we expand W into a power series of the

Page 60: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

59

four variables ( , , , )x y x yH H ρ ρ , following Sir Hamilton, let us assume that W can be

expressed by a series expansion of the form [20]

=W +W +W +W +W +…, W (0) (1) (2) (3) (4)

here W is of degree in ( ,( )n n , , )x y x yH H ρ ρ .

Using the condition of double plane symmetry, we can easily verify that only certain

combinations of these four coordinate variables can exist [14]. More precisely, only 6

double plane symmetry invariants can exist in the power series expansion. These double

plane symmetry invariants are 2 2 2 2, , ,x y xa H b H c d yρ ρ= = = = , x xe H ρ= , y yf H ρ= .

The combinations of all these six invariants will give the remaining aberrations for

different orders. It is obvious that anamorphic systems do not have W (1) , W (3 and all

other odd order terms because any combinations of are of even order

in ( ,

)

, , , , ,a b c d e f

, , )x y x yH H ρ ρ .

The order W (0 is a constant piston term and has no aperture dependence, thus it is

commonly ignored.

)

For W (2 , it will have six terms in total, among which there are two piston terms that

are ignored. Thus

)

=W (2)1 2 3 4B c B e B f B d+ + + = 2

1 2 3 42

x x x y y yB B H B H Bρ ρ ρ ρ+ + + (3-5)

These are the first-order optics terms for an anamorphic system, which define the ideal

image properties. For reference sphere centered on the ideal image point, they will all

disappear [2,28].

Page 61: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

60

ForW , it will have 21 fourth-order combinations, of which three are piston terms

and two terms are identical with other terms ( ). Thus there will be

sixteen different anamorphic primary aberration types, which can be classified into four

general types according to their aperture and field dependences

(4)

2 2,e ac f bd= =

(4) 2 21 2 3

4 5 6 7

8 9 10 11 12

13 14 15 16

{ },{ },{ },{ }, .

W D c D d D cd SphericalD ce D cf D de D df ComaticD ac D bd D bc D ad D ef Astigmatics FCD ae D bf D af D be Distortion

= + ++ + + ++ + + + + ++ + + +

It can be rewritten as

(4) 4 4 2 21 2 3

3 2 2 34 5 6 7

2 2 2 2 2 2 2 28 9 10 11 12

3 3 2 213 14 15 16

W =

.

x y x y

x x y x y x x y y y

x x y y y x x y x y x

x x y y x y x x y y

D D D

D H D H D H D H

D H D H D H D H D H H

D H D H D H H D H H

ρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

yρ ρ ρ ρ

ρ ρ ρ ρ

+ +

+ + + +

+ + + + +

+ + + +

ρ ρ (3-6a)

Equation (3-6a) is the primary wave aberration expansion for anamorphic systems,

and though are the anamorphic primary aberration coefficients. It should be

noticed that anamorphic systems are much more complex than RSOS which have five

primary (third-order) aberrations types only.

1D 16D

Considering cos , sin , cos , sin ,x y x yH H H Hφ φ ρ ρ θ ρ ρ θ= = = = we can rewrite

equation (3-6a) into its polar form. By regrouping the terms according to their field and

aperture dependences, we have

(4) 4 4 2 2 41 2 3

34 5

2 36 7

W ={ cos sin sin cos }

{ cos cos sin sin cos

cos sin cos sin sin }

D D D

D D

D D

2

3H

θ θ θ θ

φ θ φ θ θ

ρ

φ θ θ φ θ ρ

+ +

+ +

+ +

(3-6b)

Page 62: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

61

2 2 2 2 2 28 9 10

2 2 2 211 12

3 313 14

2 215 16

{ cos cos sin sin sin cos

cos sin sin cos sin cos }

{ cos cos sin sin

sin cos cos sin cos sin } .

D D D

D D H

D D

D D H 3

φ θ φ θ φ

φ θ φ φ θ θ ρ

φ θ φ θ

φ φ θ φ φ θ ρ

+ + +

+ +

+ +

+ +

θ

The typical wavefront shape of the anamorphic primary wave aberration types are

shown in Appendix A.

By taking the derivative of equation (3-6a) with respect to ,x yρ ρ while applying

equations (3-4c), we get the primary ray error expansions for anamorphic systems as

3 2 2 2

1 3 4 5 6 8

2 3 210 12 13 15 ,

(4 2 3 2 2

2 )

2

/ ' ' ,x x y x x y x y x y x x

y x x y y x x y x k

D D D H D H D H D H

D H D H H D H D H H n u

δξ ρ ρ ρ ρ ρ ρ ρ

ρ ρ

= + + + + +

+ + + +

ρ

2

' ' .y

(3-7a)

3 2 2 2

2 3 5 6 7 9

2 3 211 12 14 16 ,

(4 2 2 3 2

2 ) /y x y y x x x y y y y

x y x y x y x y y k

D D D H D H D H D H

D H D H H D H D H H n u

δη ρ ρ ρ ρ ρ ρ ρ

ρ ρ

= + + + + +

+ + + +

ρ (3-7b)

From equations (3-7), we can separate the corresponding ray errors for each

anamorphic primary aberration type, as shown below.

For Spherical Aberration-like aberration types

1 :D3

, 1

,

' ' 4' ' 0

x k x

y k

n u Dn u

δξ ρδη

⎫= ⎪⎬

= ⎪⎭ , (3-8a)

2 :D,

3, 2

' ' 0

' ' 4x k

y k y

n u

n u D

δξ

δη ρ

= ⎫⎪⎬

= ⎪⎭ , (3-8b)

3 :D2

, 3

2, 3

' ' 2

' ' 2x k x y

y k x

n u D

n u D

δξ ρ ρ

yδη ρ ρ

⎫= ⎪⎬

= ⎪⎭ . (3-8c)

For Coma-like aberration types

Page 63: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

62

4 :D2

, 4

,

' ' 3' ' 0

x k x x

y k

n u D Hn u

δξ ρδη

⎫= ⎪⎬

= ⎪⎭ , (3-8d)

5 :D, 5

2, 5

' ' 2

' 'x k y

y k y x

n u D H

n u D Hx yδξ ρ

δη ρ

= ⎫ρ ⎪⎬

= ⎪⎭, (3-8e)

6 :D2

, 6

, 6

' '' ' 2

x k x y

y k x x y

n u D Hn u D H

δξ ρ

δη ρ

⎫=

ρ⎪⎬

= ⎪⎭, (3-8f)

7 :D,

2, 7

' ' 0

' ' 3x k

y k y y

n u

n u D H

δξ

δη ρ

= ⎫⎪⎬

= ⎪⎭. (3-8g)

For Astigmatism and Field curvature-like aberration types

8 :D2

, 8

,

' ' 2' ' 0

x k x

y k

n u D Hn u

xδξ ρδη

⎫= ⎪⎬

= ⎪⎭, (3-8h)

9 :D,

2, 9

' ' 0

' ' 2x k

y k y y

n u

n u D H

δξ

δη ρ

= ⎫⎪⎬

= ⎪⎭, (3-8i)

10 :D2

, 10

,

' ' 2' ' 0

x k y

y k

n u D Hn u

xδξ ρ

δη

⎫= ⎪⎬

= ⎪⎭, (3-8j)

11 :D,

2, 11

' ' 0

' ' 2x k

y k x y

n u

n u D H

δξ

δη ρ

= ⎫⎪⎬

= ⎪⎭, (3-8k)

12 :D , 12

, 12

' '' '

x k x y

y k x y x

n u D H Hn u D H H

yδξ ρ

δη ρ

= ⎫⎪⎬= ⎪⎭

. (3-8 l )

For Distortion-like aberration types

13 :D3

, 13

,

' '' ' 0

x k x

y k

n u D Hn u

δξδη

⎫= ⎪⎬

= ⎪⎭, (3-8m)

Page 64: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

63

14 :D,

3, 14

' ' 0

' 'x k

y k y

n u

n u D H

δξ

δη

= ⎫⎪⎬

= ⎪⎭, (3-8n)

15 :D2

, 15

,

' '' ' 0

x k x y

y k

n u D H Hn u

δξ

δη

⎫= ⎪⎬

= ⎪⎭, (3-8o)

16 :D,

2, 16

' ' 0

' 'x k

y k x y

n u

n u D H H

δξ

δη

= ⎫⎪⎬

= ⎪⎭. (3-8p)

These 16 equations will serve as our models and the 16 anamorphic primary

aberration coefficients though can be derived by comparing with them, as will be

shown in chapter 5 through 8.

1D 16D

3.4 Summary

In this chapter, we constructed the aberration function for anamorphic systems with

object and stop variables, and showed that there were 16 primary aberration types,

indicating a greater complexity than RSOS. We also built up the relationship between the

primary wave aberrations and the primary ray aberrations.

Page 65: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

64

CHAPTER 4

METHOD OF ANAMORPHIC PRIMARY ABERRATIONS CALCULATION

In chapter 2, we built the first-order theory for anamorphic imaging systems. In

chapter 3, we built the anamorphic aberration function and its power series expansion; we

shown there were sixteen primary aberration types in anamorphic imaging systems; we

also built the connection between primary wave error and ray errors, for each primary

aberration type. In this chapter, we will build the actual method for deriving the sixteen

anamorphic primary aberration coefficient expressions.

We will present the total ray aberration equations for anamorphic systems in section

4.1, and we will reduce them into the anamorphic primary ray aberration equations in

sections 4.2 through 4.4.

4.1 The total ray aberration equations for anamorphic systems

In geometric optics for RSOS, there is a so-called Aldis theorem [2,22], which gives

expressions for the traverse ray aberration components 'kδξ and '

kδη of a finite (real) ray

with respect to the ideal (paraxial) image location. The idea of the Aldis theorem in

RSOS can be described as follows

1) First suppose the image system is ideal and has no error;

2) Now suppose the aberrations can be considered to arise intrinsically on a specific

surface, with this surface error producing a specific distribution of errors in the

intermediate image plane of this surface;

Page 66: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

65

3) The function of the remainder of the lenses in the system can be considered as

transferring that intermediate error distribution to the final image plane.

The same is true for other surfaces. Therefore, if we can calculate the contribution of

the initial error 'jε , then we can sum up the contributions from each surface in the final

image plane and get the total ray aberration for the system.

In the 1960s, A. Cox has found that the primary aberration coefficients for RSOS can

be deducted from the Aldis theorem [22]. However, we realize that a generalized idea can

be applied outside of RSOS, and will be able to get the primary aberration coefficients for

anamorphic systems.

Remember that in an anamorphic system, we have two sets of intermediate paraxial

image planes floating in space, namely the x-intermediate image planes and the y-

intermediate image planes, each associated with one principal section of the system.

Consider a system with k surfaces that are a mixture of spherical surfaces and double

curvature surfaces. Let ( , ,j j jNL M ) be the direction cosines of a finite (real) ray and let

( , ,j j jx y z ) be the coordinates of the point of incidence on surface j of this ray. For the

chosen object field ( ,x yH H ), let 0, jξ be the ideal x-image height in the x-intermediate

image plane of surface j . Similarly, let 0, jη be the ideal y-image height in the y-

intermediate image plane of surface j . And let ,x jA and ,y jA be the refraction invariants

associate with the corresponding x and y marginal rays, as defined in chapter 2.11.

Notice that the Lagrange invariant is calculated with the paraxial chief ray data from

the maximum field point. For an object point at fractional field, we should use the

Page 67: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

66

fractional Lagrange invariant instead. For the anamorphic system described above, we

have the x-fractional Lagrange invariant , 0,x x j x jH n u jψ ξ= , and the y-fractional Lagrange

invariant as , 0,y y j y jH n u jψ η= .

Now let us just consider surface j of the system, as shown in Figure 4-1 below.

Suppose a finite skew ray from the fractional object field ( ,x yH H ), which intersects the

refraction surface at point ( , ,x y z ), then crosses the x-intermediate image plane at

point ( , )ξ and the y-intermediate image plane at point ( , )η . Here symbol “ ” means

we do not care the value of the corresponding coordinates.

Figure 4-1 Refraction on surface j

From equations (2-6) we have the ray transfer equations

( xL )x l zN

ξ = + − , (4-1a)

( yMy lN

η )z= + − , (4-1b)

Page 68: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

67

where xl and yl are the distance from the vertex of the refraction surface to the x-

intermediate image plane and y-intermediate image plane, respectively, and in

general x yl l≠ .

From equation (4-1a), multiplying through by , we get xnu N

x x x xnu N nu Nx nu Ll nu Lzxξ = + − . (4-2)

By applying equation (2-41a), we can rewrite equation (4-2) as

( ) ( )x x x x x x x xnu N A nh c Nx A nh c Lz nu Llxξ = − − − + . (4-3)

From the fraction x-Lagrange invariant 0x x xnu Hξ ψ= , we have

0x x xnu N NHξ ψ= , (4-4)

here 0ξ is the ideal x-image height at the x-intermediate image plane.

Subtracting equation (4-4) from equation (4-3), we have

( ) ( )

( ) (x x x x x x x

x x x x x x x x x

nu N A xN zL nh c xN zL nh L H NA x H N A zL h c xnN h c z h nL) ,

δξ ψψ

= − − − − −= − − − + −

(4-5a)

here we have made use of the paraxial definition /x x xu h l= − . And by definition

0δξ ξ ξ= − is the x-ray error in this space. Similarly, we can find

( ) ( )

( ) (y y y y y y y

y y y y y y y y y

nu N A yN zM nh c yN zM nh M H N

) .A y H N A zM h c ynN h c z h nM

δη ψ

ψ

= − − − − −

= − − − + − (4-5b)

Till this step, we have got the corresponding specific distribution of errors in the

corresponding intermediate image planes of surface j . We now need to transfer these

error distributions through the system to the final image space.

Page 69: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

68

By applying the refraction operator Δ on equation (4-5a), we get the increment of

this quantity on refraction as

{ } {( ) ( ) }

( ) ( )x x x x x x x x x x

x x x x x x x x x

nu N A x H N A zL h c xnN h c z h nLA x H N A z L h c x nN h c z h nL.

δξ ψψ

Δ = Δ − − − + −= − Δ − Δ − Δ + − Δ

(4-6a)

Notice that all refraction constants on this surface come out of the refraction operator

because they do not change their value on refraction. Similarly, for the y component, we

will get

{ } ( ) ( )y y y y y y y y y ynu N A y H N A z M h c y nN h c z h nMδη ψΔ = − Δ − Δ − Δ + − Δ . (4-6b)

From equation (2-11b), we can find

nL nNαγ

Δ = Δ , (4-7a)

nM nNβγ

Δ = Δ . (4-7b)

Substituting equation (4-7a) into equation (4-6a), we get

{ } ( ) [( ) ]x x x x x x x x x xnu N A x H N A z L h c z h h c x nNαδξ ψγ

Δ = − Δ − Δ + − − Δ . (4-8)

Notice that equation (4-8) is valid for every intermediate space. Also notice that on

surface j , '{ } { } { }x j x j xnu N nu N nu N jδξ δξΔ = − δξ } and 1{ }' {x xnu N nu N jδξ δξ += . Thus

we are able to sum equation (4-8) through all surfaces to the final image space of the

system. By taking into account the fact that most likely in object space, 1 1xn u N 1δξ and

1 ,1 1 1yn u N δη are zero because the object have no aberration, we get the expression for the

x-component of the system ray aberration to be

Page 70: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

69

' ' ' ', ,

1 1

, , , , ,1

( )

[( ) ] .

k k

k x k k k x j j j j j x x jj j

kj

x j x j j x j x j x j j j jj j

n u N A x N z L H N

h c z h h c x n N

δξ ψ

αγ

= =

=

= Δ − Δ − Δ

+ − − Δ

∑ ∑

∑ (4-9a)

Similarly, we get the expression for the y-component of system ray aberration to be

' ' ' ', ,

1 1

, , , , ,1

( )

[( ) ] .

k k

k y k k k y j j j j j y y jj j

kj

y j y j j y j y j y j j j jj j

n u N A y N z M H N

h c z h h c y n N

δη ψ

βγ

= =

=

= Δ − Δ − Δ

+ − − Δ

∑ ∑

∑ (4-9b)

Equations (4-9a) and (4-9b) are the anamorphic total ray aberration equations.

Notice that an RSOS made from spherical surfaces is a special case of an anamorphic

system with the anamorphic ratio being one, thus equation (4-9a) and (4-9b) are valid for

RSOS also. We can easily verify that in the RSOS case these equations will be in a

simpler form

' ' ' ', ,

1 1

( )k k

k x k k k x j j j j j x x jj j

n u N A x N z L H Nδξ ψ= =

= Δ − Δ − Δ∑ ∑ , (4-9c)

' ' ' ', ,

1 1

( )k k

k y k k k y j j j j j y y jj j

n u N A y N z M H Nδη ψ= =

= Δ − Δ − Δ∑ ∑ . (4-9d)

Remember that equations (4-9) are total ray error equations, in that they include ray

errors of all orders. For primary aberration derivation purpose, we only need the third-

order ray error (corresponding to the fourth-order wave error), thus we need to reduce the

anamorphic total ray aberration equations into their third-order equivalents—the

anamorphic primary ray aberration equations.

Page 71: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

70

4.2 Preparation for the anamorphic primary ray aberration equations deduction

We now need to find ,α βγ γ

as shown up in the basic equations (4-9a)-(4-9b). We can

rewrite the general double curvature surface sag equation (2-18) as

2 2 4 2 2 4

3 3 33 4 5

1 1 2( , , ) ( ) ( ) 02 8x y

x y x x y yF x y z zr r r r r

= − + − + + = . (4-10)

Following the method described in chapter 2, section 3, we find

3 2

33 42 2x

F x x xy3x r r r

∂= − − −

∂,

3 2

35 42 2y

F y y xy r r r

∂= − − −

∂ 3y ,

1Fz

∂=

∂.

We define the quantity1

2 2 2{( ) ( ) ( ) }F F FGx y z

∂ ∂ ∂= + +

∂ ∂ ∂2 . From the direction cosines of the

surface normal equation (2-21), we have

( / ) 1F zG G

γ ∂ ∂= = .

This gives ( / ) / ( /F x G F x)α γ= ∂ ∂ = ∂ ∂ and ( / ) / ( /F y G F y)β γ= ∂ ∂ = ∂ ∂ , which result in

3 2

33 4

(2 2x

F x x xy3 )

x r r rαγ

∂= = − + +∂

, (4-11a)

3 2

35 4

(2 2y

F y y xy r r r

βγ

∂= = − + +∂ 3 )y . (4-11b)

These equations are accurate up to the third-order.

Page 72: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

71

For the primary aberration calculation, we can take , , ,j j j jx y L M as their paraxial

equivalents [2,28]. Hence we have

1j jx x= , (4-12a)

1j jy y= , (4-12b)

1 ,j j xL L u j= = , (4-12c)

1 ,j j y jM M u= = , (4-12d)

here the subscript 1 denotes the above quantities are of order one in paraxial ray heights

and angles. The double bar has the same meaning as in chapter 2, which shows we are

dealing with an arbitrary paraxial ray other than the known marginal and chief rays in the

two associated RSOS.

Equation (2-4) can now be rewritten as

2 2

1 ... 12

j jj j

L MN Nδ

+= − + = − ,

here

2 2

...2

j jj

L MNδ

+= + .

By putting equations (4-12c) and (4-12d) into jNδ and ignoring all terms higher than

third-order (because they only contribute to higher order aberrations), we get

2 2

12 2

1j jj j

L MN Nδ δ

+= = . (4-12e)

Thus we have

2 1 2j jN N N jδ= = − , (4-13a)

Page 73: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

72

And

2 2j j j jn N n n N jδ= − , (4-13b)

here subscript 2 denotes that quantity is of second-order in ray heights and angles.

By applying equation (4-13a), if we move the on the left-hand side to the right-

hand side of equations (4-9), up to third-order accuracy, we will have

'kN

'2' '

2

1 1 11 k

k k

NN N

δδ

= = +−

. (4-13c)

By putting equations (4-12a) and (4-12b) into the general double curvature surface

sag equation (2-18) and ignoring all terms higher than order three, we get

2 2

1 12

, ,

1 (2

j jj j )

x j y j

x yz z

r r= = + . (4-13d)

By putting equations (4-12a) and (4-12b) into equations (4-11a) and (4-11b), and

ignoring all terms higher than third-order, we get

3 21 1 1 1

3 3, 3, 4,

3 21 1 1

3, 3, 4,

1 3

1 3

( )2 2

( ) (2 2

,

j j j j j

j x j j j

13 )j j j j

x j j j

j j

j j

x x x yr r r

x x x yr r r

αγ

α αγ γ

= − + +

= − − +

= +

(4-14a)

here3 2

1 1 3 1 13

1 , 3 3, 4,

( ), (2 2

13 )j j j j j j

j x j j j j

x x x yr r

α αγ γ

= − = − +r

. Subscript 3 denotes that quantity is of third-

order in ray heights and angles.

Similarly, we can find

Page 74: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

73

1

1 3

3j j j

j j j

β β βγ γ γ

= + , (4-14b)

here3 2

1 1 3 1 13 3

1 , 3 5, 4,

( ), (2 2

1 )j j j j j j

j y j j j j

y y xr r

β βγ γ

= − = − +y

r.

Because the third summation in the R.H.S of equations (4-9a)-(4-9b) is quite lengthy,

let us treat them separately. For the one in equation (4-9a)

, , , , ,1

[( ) ]k

jx j x j j x j x j x j j j j

j j

h c z h h c x n Nαγ=

− − Δ∑ ,

by putting equations (4-13) though (4-14) into it and ignoring all terms higher than third-

order, also using the relation ,1/ ,x j xr c j= , we get

, , , , ,1

1 3, , , , ,

1 , 3

1 3 ' ' ', , 2 , 2 2

1 , 3

' 2, , 2

1

[( ) ]

[( )( ) ]

[ ( ) ][( ) ( )]

( )(

kj

x j x j j x j x j x j j j jj j

kj j

x j x j j x j x j x j j j jj x j j

kj j

x j x j j x j j j j j j jj x j j

k

x j j j x j j jj

h c z h h c x n N

xh c z h h c x n N

rx

h c z h n n n N n Nr

h n n c z x

αγ

αγ

αδ δ

γ

=

=

=

=

− − Δ

= − − + − Δ

= − − − − −

= − −

∑ 31

3

).j

j

αγ

+

(4-15a)

Similarly, we can simplify the third summation in equation (4-9b) as

, , , , ,

1

3' 2, , 2 1

1 3

[( ) ]

( )( ).

kj

y j y j j y j y j y j j j jj j

kj

y j j j y j j jj j

h c z h h c y n N

h n n c z y

βγ

βγ

=

=

− − Δ

= − − +

∑ (4-15b)

Again, these equations are accurate up to the third-order.

Page 75: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

74

4.3 The anamorphic primary ray aberration calculation-part one

With all approximations in place, we can now reduce the anamorphic total ray

aberration equations (4-9) to their third-order equivalents—the anamorphic primary ray

aberration equations.

Let us put equations (4-12) through (4-15) into the R.H.S of equations (4-9a)-(4-9b),

by ignoring all terms higher than order three, we get

' ' ', 3 , 1 2 2 1

1

322 , , 2 1

1 1 3

( )

( )

k

k x k k x j j j j jj

k kj ,x x j x j x j j j j

j j j

n u A x N z L

H N h c z x

δξ δ

αψ δ

γ

=

= =

= − Δ + Δ

+ Δ − + Δ

∑ ∑ n (4-16a)

' ' ', 3 , 1 2 2 1

1

322 , , 2 1

1 1 3

( )

( )

k

k y k k y j j j j jj

k kj

y y j y j y j j j jj j j

n u A y N z M

H N h c z y

δη δ

βψ δ

γ

=

= =

= − Δ + Δ

+ Δ − + Δ

∑ ∑ .n (4-16b)

By pulling the refraction operator out of the expressions, we finally obtain

' ' ', 3 , 1 2 2 1

1

322 , , 2 1

1 1 3

[ ( )

( )

k

k x k k x j j j j jj

k kj ],x x j x j x j j j j

j j j

n u A x N z L

H N h c z x n

δξ δ

αψ δ

γ

=

= =

= Δ − +

+ − +

∑ ∑ (4-17a)

' ' ', 3 , 1 2 2 1

1

322 , , 2 1

1 1 3

[ ( )

( )

k

k y k k y j j j j jj

k kj

y y j y j y j j j jj j j

n u A y N z M

H N h c z y

δη δ

βψ δ

γ

=

= =

= Δ − +

+ − +

∑ ∑ ].n (4-17b)

These two equations are the anamorphic primary ray aberration equations, and they will

serve as our basic equations for the anamorphic aberration coefficients derivation.

Page 76: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

75

Again, equations (4-17a) and (4-17b) are valid for RSOS too. We can easily verify

that in RSOS case, these equations will be in a simpler form

, (4-18a) ' ' ', 3 , 1 2 2 1 2

1 1

[ ( )k k

k x k k x j j j j j x x jj j

n u A x N z L H Nδξ δ ψ δ= =

= Δ − + +∑ ]∑

]

3

. (4-18b) ' ' ', 3 , 1 2 2 1 2

1 1

[ ( )k k

k y k k y j j j j j y y jj j

n u A y N z M H Nδη δ ψ δ= =

= Δ − + +∑ ∑

4.4 The anamorphic primary ray aberration calculation-part two

Now what we need to do is to replace 1 1 1 1 2 2 3 3 3, , , , , , / , /j j j j j j j j j jx y L M N zδ α γ β γ

with the four known paraxial marginal and chief rays’ tracing data in the two associated

RSOS, together with aperture and field variables.

Since 1 1 1, , , 1j j j jx y L M are all paraxial quantities associated with an arbitrary ray, as

indicated by equations (4-12a) through (4-12d), we can make use of equations (2-40) to

write them as

1 ,j x x j x x , jx h H hρ= + , (4-19a)

1 ,j y y j y yy h H hρ= + , j , (4-19b)

1 , ,j x x j x x jL u H uρ= + , (4-19c)

1 , ,j y y j y y jM u H uρ= + , (4-19d)

By putting equations (4-19c)-(4-19d) into equation (4-12e), we get

2 2

1 1 22 , , ,

1 [( ) ( ) ]2 2

j jj x x j x x j y y j

L MN u H u uδ ρ ρ

+= = + + + 2

,y y jH u . (4-19e)

Page 77: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

76

By putting equations (4-19a) and (4-19b) into the corresponding expressions of

2 3 3 3, / , / 3j j j j jz α γ β γ , we know that all parameters in equations (4-17) can indeed be

written as combinations of aperture and field variables. This, together with the ray tracing

data of the marginal and chief rays in the two associated RSOS, allows us to actually

calculate the 16 primary aberration coefficients for anamorphic systems.

4.5 Summary

In this chapter, by applying the generalized Aldis idea onto anamorphic imaging

systems, we built up the anamorphic total ray aberration equations. We then reduced

these equations into their third-order equivalents: The anamorphic primary ray aberration

equations. We then wrote all parameters in the anamorphic primary ray aberration

equations in terms of the paraxial ray trace data in the two associated RSOS, together

with field and aperture variables.

With these steps, we lay the groundwork of actually obtaining the 16 primary

aberration coefficients for different types of anamorphic systems, as will be described in

the chapters to follow.

Page 78: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

77

CHAPTER 5

PRIMARY ABERRATION THEORY FOR PARALLEL CYLINDRICAL

ANAMORPHIC ATTACHMENT SYSTEMS

From chapters 2 through 4, we have detailed the general method of deriving the

anamorphic primary aberration coefficient expressions for any anamorphic system types

with double plane symmetry.

We will want to consider one very important difference between RSOS and

anamorphic systems. In RSOS, due to the rotational symmetry, all surfaces inside the

systems will be spherical or spherical with even aspheric departure, thus the primary

aberration coefficient expressions are basically in the same form for all RSOS. In other

words, there are different rotation symmetric systems but they will all be the same type.

For anamorphic systems, however, there are many different surface types which can

be included in the systems, such as cylindrical, toroidal, ellipsoidal, etc. These different

surface types all possess of double plane symmetry but will have different surface

equations. From paraxial image formation perspective, these different surfaces types are

the same in the paraxial region. But from aberration perspective, anamorphic systems

made from different surface types will have different primary aberration coefficient

expressions, hence they need to be treated as different anamorphic system types, namely

cylindrical anamorphic systems, toroidal anamorphic systems, etc.

Under this complication, we are facing a choice here. We can choose to group all

types of double curvature surfaces together and write them into a general surface type and

Page 79: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

78

then come up with a set of lengthy but generally applicable anamorphic primary

aberration coefficient expressions. The problem with this choice is that the resulting

analytical expressions might be so complex that they actually obscure any insight of the

results. Or we can explore different types of anamorphic systems separately, with

increasing complexity. In this manner, for some simple yet important anamorphic system

types, the analytical results will be much easier to follow and can provide much better

design insight.

In this work, we will explore different types of anamorphic systems with increasing

complexity. We will develop the primary aberration coefficients for parallel cylindrical

anamorphic systems, cross cylindrical anamorphic systems, and toroidal anamorphic

systems separately in chapters 5 through 7. In chapter 8, we will present the general

analytical expressions which are applicable to any type of anamorphic system. This will

allow the reader to reference the specific chapters which will be of interest and find the

corresponding primary aberration coefficients without being lost in the math details.

In the following chapters, since the method we developed is generally valid for any

type of anamorphic system, its application onto different anamorphic system types is

quite mechanical which may make the derivation procedures seem somewhat boring. But

this is actually an advantage of our method since it eliminates the need for exceptions to

the general method because of special differences in the systems. To illustrate this

important point, in the derivation process that follows, we will purposely use the same

structure for each chapter:

Page 80: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

79

a) We will first apply our method to find the primary ray errors for the specified

anamorphic system type under study;

b) Once we get the primary ray errors, by comparing with the relationship between

the primary wave error and ray errors obtained in chapter 3.3, we can immediately

get the primary wave aberration coefficients for the anamorphic system under

study;

c) We then simplify the results to make them in a form as similar to the Seidel

aberrations for RSOS as possible.

Following this pattern for the rest of this chapter, we will present the primary ray

aberration equations for cylindrical anamorphic systems in section 5.1, the primary ray

aberration coefficients for parallel cylindrical anamorphic attachment systems in section

5.2, and the primary wave aberration coefficients for parallel cylindrical anamorphic

attachment systems and their simplification in sections 5.3 and 5.4.

5.1 The primary ray aberration equations for cylindrical anamorphic systems

From a manufacturing perspective, the most easily made double curvature surfaces

are cylindrical surfaces with one radius of curvature equal to infinity, so it is not

unexpected that the most commonly used anamorphic system are made from cylindrical

lens. For the same reason, our first research subject is an anamorphic system made from

cylindrical lenses.

For a cylindrical refracting surface with yr = ∞ , the surface sag equation with a

fourth-order approximation is

Page 81: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

80

2 4

31 12 8x x

x xzr r

= + . (5-1a)

For this case, equation (4-13d) becomes

22 ,

12 1j x j jz c x= . (5-1b)

Since , from equations (4-14a) and (4-14b), we have 3 4 5,xr r r r= = = ∞

3 3, 1

3

12

j 3x j j

j

c xαγ

= − , (5-1c)

3

3

0j

j

βγ

= . (5-1d)

For a cylindrical refracting surface with xr = ∞ , the surface sag equation is similarly

2 4

31 12 8y y

yzr r

= +y . (5-2a)

Now equation (4-13d) becomes

22 ,

12 1j y j jz c y= . (5-2b)

And in this case 3 4 5, yr r r r= = ∞ = , from equations (4-14a) and (4-14b) we have

3

3

0j

j

αγ

= , (5-2c)

3 3 3, 1

3

12

jy j j

j

c yβγ

= − . (5-2d)

In both cases, we can verify that the general anamorphic primary ray aberration

equations (4-17a) and (4-17b) can be reduced into

Page 82: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

81

, (5-3a) ' ' ', 3 , 1 2 2 1 2

1 1

[ ( )k k

k x k k x j j j j j x x jj j

n u A x N z L H Nδξ δ ψ δ= =

= Δ − + +∑ ]∑

] . (5-3b) ' ' ', 3 , 1 2 2 1 2

1 1

[ ( )k k

k y k k y j j j j j y y jj j

n u A y N z M H Nδη δ ψ δ= =

= Δ − + +∑ ∑

Thus, equations (5-3) are the primary ray aberration equations for any kind of cylindrical

anamorphic system.

For cylindrical anamorphic systems, we encounter two configurations which should

be considered separately because of their importance.

5.2 Primary ray aberration coefficients for parallel cylindrical anamorphic systems

The first configuration is an anamorphic attachment made from parallel cylindrical

lenses, which is often combined with a standard optical imaging system for spherical

power [13]. Here parallel means the generation lines of the cylindrical surfaces are

parallel to each other. An example of this configuration is shown in Figure 5-1 below.

Figure 5-1 A parallel cylindrical anamorphic attachment system

For this kind of configuration, rays will behave differently depending on how they

stay in the system, as shown in Figure 5-2 below. In Figure 5-2, rays that lie in the y-z

Page 83: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

82

symmetry plane will pass through the cylindrical attachment as if the system consists

only of parallel plates. Rays staying in the x-z symmetry plane will pass through the

attachment as if the system is an x-RSOS. Notice that there is no unique Gaussian image

plane in each intermediate space because the Gaussian image planes in both principal

sections normally do not coincide. The same is true for intermediate pupils.

Figure 5-2 The y-z (top) and x-z (bottom) symmetry planes of a parallel cylindrical

attachment system

Rays that do not travel in above symmetry planes will be skew rays and their passage

through the system can be described by the three-dimensional ray refraction and transfer

equations, as described in chapter 2.

Suppose we have for any cylindrical surface, 0x jc = j , by putting equation (4-19b)

into equation (5-2b), we get

22 , 1 , , ,

1 1 (2 2

2)j y j j y j y y j y y jz c y c h H hρ= = + . (5-4)

By putting equation (5-4) together with equations (4-19) into equations (5-3a) and (5-3b),

we get

Page 84: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

83

' ' ', 3 , 1 2 2 1 2

1 1

2 2 2 2, , , , , , ,

1

2 2 2 2, , , , , , ,

1

,1

[ ( ) ]

1 [( )( 2 )2

1 [( )( 2 )2

1 [2

k k

k x k k x j j j j j x x jj j

k

x j x x j x x j x x j x x j x x x j x jj

k

x j x x j x x j y y j y y j y y y j y jj

k

x jj

n u A x N z L H N

A h H h u H u H u u

A h H h u H u H u u

A c

δξ δ ψ δ

ρ ρ ρ

ρ ρ ρ

= =

=

=

=

= Δ − + +

= − + Δ + Δ + Δ

− + Δ + Δ + Δ

∑ ∑

∑ 2 2 2 2, , , , , , ,

2 2 2 2, , , ,

1

2 2 2 2, , , ,

1

( 2 )( )]

1 ( 2 )2

1 ( 2 ),2

y j y j y y j y j y y y j y x x j x x j

k

x x x x j x x j x x x j x jj

k

x x y y j y y j y y y j y jj

h h h H h H u H u

H u H u H u u

H u H u H u u

ρ ρ ρ

ψ ρ ρ

ψ ρ ρ

=

=

+ + Δ +

+ Δ + Δ + Δ

+ Δ + Δ + Δ

Δ (5-5a)

And,

' ' ', 3 , 1 2 2 1 2

1 1

2 2 2 2, , , , , , ,

1

2 2 2 2, , , , , , ,

1

,1

[ ( ) ]

1 [( )( 2 )2

1 [( )( 2 )2

1 [2

k k

k y k k y j j j j j y y jj j

k

y j y y j y y j x x j x x j x x x j x jj

k

y j y y j y y j y y j y y j y y y j y jj

k

y jj

n u A y N z M H N

A h H h u H u H u u

A h H h u H u H u u

A c

δη δ ψ δ

ρ ρ ρ

ρ ρ ρ

= =

=

=

=

= Δ − + +

= − + Δ + Δ + Δ

− + Δ + Δ + Δ

∑ ∑

∑ 2 2 2 2, , , , , , ,

2 2 2 2, , , ,

1

2 2 2 2, , , ,

1

( 2 )( )]

1 ( 2 )2

1 ( 2 ).2

y j y j y y j y j y y y j y y y j y y j

k

y y x x j x x j x x x j x jj

k

y y y y j y y j y y y j y jj

h h h H h H u H u

H u H u H u u

H u H u H u u

ρ ρ ρ

ψ ρ ρ

ψ ρ ρ

=

=

+ + Δ +

+ Δ + Δ + Δ

+ Δ + Δ + Δ

Δ (5-5b)

These two equations are quite lengthy, but they can actually be expanded and

regrouped according to their fields and aperture dependences. Rewriting them in a form

parallel to equations (3-8), we can obtain the 16 anamorphic primary aberration

Page 85: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

84

coefficient expressions from general anamorphic aberration theory. After the regrouping,

we get the corresponding primary ray aberration terms as:

For Spherical Aberration-like aberration types

1 :D' ' ' 2 3

, 3 , , ,1

' ' ', 3

1 ( )2

0

k

k x k k x j x j x j xj

k y k k

n u A h u

n u

δξ ρ

δη=

⎫= − Δ ⎪

⎬⎪= ⎭

∑ (5-6a)

2 :D

' ' ', 3

' ' ' 2 2 3, 3 , , , , , ,

1

0

1 [ ( )]2

k x k k

k

k y k k y j y j y j y j y j y j yj

n u

n u A h u c h u

δξ

δη ρ=

⎫=⎪⎬

= − Δ + Δ ⎪⎭

∑ (5-6b)

3 :D

' ' ' 2 2 2, , , , , , ,

1

' ' ' 2 2, , , ,

1

1 [ ( )]2

1 ( )2

k

k x k k x j x j y j y j y j x j x yj

k

k y k k y j y j x j x yj

n u A h u c h u

n u A h u

δξ ρ ρ

δη ρ ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎬⎪= − Δ⎪⎭

∑ (5-6c)

For Coma-like aberration types

4 :D' ' ' 2 2 2

, 3 , , , , , , ,1

' ' ', 3

1 [ ( 2 ) ]2

0

k

k x k k x j x j x j x j x j x j x x j x xj

k y k k

n u A h u h u u u H

n u

δξ ψ ρ

δη=

⎫= − Δ + Δ − Δ ⎪

⎬⎪= ⎭

∑ (5-6d)

5 :D

' ' ', 3 , , , , , , , ,

1

' ' ' 2 2 2, 3 , , , ,

1

( )

1 ( )2

k

k x k k x j x j y j y j y j y j y j x j y x yj

k

k y k k y j y j x j y x j y xj

n u A h u u c h h u H

n u A h u u H

δξ ρ ρ

δη ψ ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎬⎪= − Δ − Δ⎪⎭

∑ (5-6e)

6 :D

' ' ' 2 2 2 2, 3 , , , , , , ,

1

' ' ', 3 , , , ,

1

1 [ ( ) ]2

( )

k

k x k k x j x j y j y j y j x j x y j x yj

k

k y k k y j y j x j x j x x yj

n u A h u c h u u H

n u A h u u H

δξ ρ

δη ρ ρ

=

=

⎫= − Δ + Δ −Ψ Δ ⎪

⎪⎬⎪= − Δ⎪⎭

∑ (5-6f)

Page 86: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

85

7 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

2 2, , , , , , , ,

0

1 [ (2

2 2 ) ]

k x k k

k

k y k k y j y j y j y j y j y jj

y j y j y j y j y j y j y j y y j y y

n u

n u A h u c h u

h u u c h h u u H

δξ

δη

ψ ρ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ Δ + Δ − Δ ⎭

∑ (5-6g)

For Astigmatism and Field curvature-like aberration types

8 :D' ' ' 2 2

, 3 , , , , , , , ,1

' ' ', 3

1 [ ( 2 ) 2 ]2

0

k

k x k k x j x j x j x j x j x j x x j x j x xj

k y k k

n u A h u h u u u u H

n u

δξ ψ

δη=

⎫= − Δ + Δ − Δ ρ ⎪

⎬⎪= ⎭

∑ (5-6h)

9 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

2, , , , , , , , ,

0

1 [ (2

2 2 ) 2

k x k k

k

k y k k y j y j y j y j y j y jj

y j y j y j y j y j y j y j y y j y j y y

n u

n u A h u c h u

c h h u h u u u u H

δξ

δη

]ψ ρ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ Δ + Δ − Δ ⎭

∑ (5-6i)

10 :D' ' ' 2 2 2

, 3 , , , , , ,1

' ' ', 3

1 [ ( )]2

0

k

k x k k x j x j y j y j y j x j y xj

k y k k

n u A h u c h u H

n u

δξ ρ

δη=

⎫= − Δ + Δ ⎪

⎬⎪= ⎭

∑ (5-6j)

11 :D

' ' ', 3

' ' ' 2 2, 3 , , ,

1

0

1 ( )2

k x k k

k

k y k k y j y j x j x yj

n u

n u A h u H

δξ

δη ρ=

⎫=⎪⎬

= − Δ ⎪⎭

∑ (5-6k)

12 :D

' ' ', 3 , , , , , , , ,

1

, ,

' ' ', 3 , , , , , ,

1

[ ( )

]

( )

k

k x k k x j x j y j y j y j y j y j x jj

x y j y j x y y

k

k y k k y j y j x j x j y x j x j x y xj

n u A h u u c h h u

u u H H

n u A h u u u u H H

δξ

ψ ρ

δη ψ ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎪− Δ ⎬⎪⎪= − Δ − Δ⎪⎭

(5-6 l )

For Distortion-like aberration types

Page 87: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

86

13 :D' ' ' 2 2 3

, 3 , , , ,1

' ' ', 3

1 ( )2

0

k

k x k k x j x j x j x x j xj

k y k k

n u A h u u H

n u

δξ ψ

δη=

⎫= − Δ − Δ ⎪

⎬⎪= ⎭

∑ (5-6m)

14 :D

' ' ', 3

' ' ' 2 2 2 3, 3 , , , , , , ,

1

0

1 [ ( ) ]2

k x k k

k

k y k k y j y j y j y j y j y j y y j yj

n u

n u A h u c h u u H

δξ

δη ψ=

⎫=⎪⎬

= − Δ + Δ − Δ ⎪⎭

∑ (5-6n)

15 :D' ' ' 2 2 2 2

, 3 , , , , , , ,1

' ' ', 3

1 [ ( ) ]2

0

k

k x k k x j x j y j y j y j x j x y j x yj

k y k k

n u A h u c h u u H H

n u

δξ ψ

δη=

⎫= − Δ + Δ − Δ ⎪

⎬⎪= ⎭

∑ (5-6o)

16 :D

' ' ', 3

' ' ' 2 2 2, 3 , , , ,

1

0

1 ( )2

k x k k

k

k y k k y j y j x j y x j x yj

n u

n u A h u u H H

δξ

δη ψ=

⎫=⎪⎬

= − Δ − Δ ⎪⎭

∑ (5-6p)

In all primary aberration types listed above, we notice and will

contribute to ray error in both x and y directions at any surface

3 5, ,D D D6 12D

j . For any single surface,

the coefficients for x-component of ray error and y-component of ray error may not be

equal, but the system aberration coefficients that come from summation through all

surfaces will be equal.

5.3 Primary wave aberration coefficients for parallel cylindrical anamorphic systems

By comparing equations (5-6) with equations (3-8), we immediately get the primary

wave aberration coefficients though as 1D 16D

21 , ,

1

18

k

,x j x j x jj

D A h=

= − Δ∑ u (5-7a)

Page 88: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

87

2 22 , , , , ,

1

1 (8

k

y j y j y j y j y j y jj

D A h u c h=

= − Δ + Δ∑ , )u (5-7b)

23 , ,

1

14

k

y j y j x jj

D A h=

= − Δ∑ ,u (5-7c)

2 24 , , , , , ,

1

1 [ ( 2 )6

k

, ]x j x j x j x j x j x j x x jj

D A h u h u u uψ=

= − Δ + Δ − Δ∑ (5-7d)

2 25 , , ,

1

1 (2

k

y j y j x j y x jj

D A h u uψ=

= − Δ − Δ∑ , ) (5-7e)

6 , ,1

12

k

y j y j x j x jj

D A h u=

= − Δ∑ , ,u (5-7f)

2 2

7 , , , , , , ,1

2, , , , ,

1 [ ( 26

2 ) ]

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j

D A h u c h u h u

c h h u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ , ,u (5-7g)

28 , , , , , ,

1

1 [ ( 2 ) 24

k

, , ]x j x j x j x j x j x j x x j x jj

D A h u h u u uψ=

= − Δ + Δ − Δ∑ u (5-7h)

2 2

9 , , , , , , ,1

, , , , , ,

1 [ ( 24

2 ) 2 ]

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j

D A h u c h u h u

c h h u u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ , ,u (5-7i)

2 210 , , , , , ,

1

1 (4

k

)x j x j y j y j y j x jj

D A h u c h=

= − Δ + Δ∑ u (5-7j)

211 , , ,

1

14

k

y j y j x jj

D A h=

= − Δ∑ u (5-7k)

12 , , , , , ,1

( )k

y j y j x j x j y x j x jj

D A h u u uψ=

= − Δ − Δ∑ u (5-7 l )

2 213 , , , ,

1

1 (2

k

)x j x j x j x x jj

D A h u ψ=

= − Δ − Δ∑ u (5-7m)

Page 89: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

88

2 214 , , , , , , ,

1

1 [ ( )2

k

y j y j y j y j y j y j y y jj

D A h u c h u ψ=

= − Δ + Δ − Δ∑ 2 ]u (5-7n)

2 215 , , , , , , ,

1

1 [ ( )2

k2 ]x j x j y j y j y j x j x y j

j

D A h u c h u ψ=

= − Δ + Δ − Δ∑ u (5-7o)

2 216 , , , ,

1

1 (2

k

y j y j x j y x jj

D A h u ψ=

= − Δ − Δ∑ )u (5-7p)

Again, Notice that for , we can either choose the coefficient in the x-

component of the ray error expression or y-component of the ray error expression. Their

numerical values may differ for any single surface

3 5 6 12, , ,D D D D

j , but the summation in the final

image space will be the same. In our treatment above, the coefficients in y-component of

the ray error expression were arbitrarily chosen.

5.3 Simplification of the results

Now we have all the primary wave aberration coefficients for parallel cylindrical

anamorphic systems in terms of the paraxial marginal and chief ray trace data in both

associated RSOS. Next let us use the corresponding paraxial definitions from chapter

2.11 to simplify the results so that we can rewrite equations (5-7) in a form as similar to

the Seidel aberrations for RSOS as possible. For the current system, because , 0x jc = , we

have

, , , , ,x j j x j j x j x j j x jA n u n h c n u= + = ,

, , , , ,x j j x j j x j x j j x jA n u n h c n u= + = ,

, , , ,y j j y j j y j y jA n u n h c= + ,

Page 90: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

89

, , ,y j j y j j y j y j,A n u n h c= + ,

, , , , , , , ,( )x j x j x j x j x j x j x j x j x jn h u h u A h A hΨ = − = − ,

, , , , , , , ,( )y j y j y j y j y j y j y j y j y jn h u h u A h A hΨ = − = − .

Notice that all these parameters are refraction invariants. From the definition of refraction

constants, we know that whenever the refraction operator Δ acts on a refraction constant,

the result will be zero. We will make use of this convenient property in our simplification

process below.

We also define

,,

,,

,

.

x jx j

j

y jy j

j

cP

nc

Pn

= Δ

= Δ

Simplifying equations (5-7) using the above paraxial definitions, for , we find 1D

21 , ,

1

,2, ,

1

18

181 .8

k

,x j x j x jj

kx j

x j x jj j

Ix

D A h

uA h

n

S

=

=

= − Δ

= − Δ

= −

u

(5-8a)

IxS means the result is in the same form as the Seidel spherical aberration for x-RSOS,

where all spherical surfaces in the x-RSOS are plane surfaces with radii of curvature

equal to infinity.

For , we find 2D

Page 91: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

90

2 22 , , , , ,

1

, , , , , ,1

,, , , , ,

1

,2, ,

1

1 ( )8

1 ( )8

1 ( )8

181 .8

k

y j y j y j y j y j y jj

k

y j y j y j y j y j y jj

ky j

y j y j j y j j y j y jj j

ky j

y j y jj j

Iy

D A h u c h u

A h u u c h

uA h n u n c h

nu

A hn

S

=

=

=

=

= − Δ + Δ

= − Δ +

= − Δ +

= − Δ

= −

,

(5-8b)

Similarly, here IyS is the Seidel spherical aberration for y-RSOS.

For , we find 3D

23 , , ,

1

,, , ,

1

14

1 .4

k

y j y j x jj

kx j

x j y j y jj j

D A h u

uA A h

n

=

=

= − Δ

= − Δ

∑ (5-8c)

For , we find 4D

2 24 , , , , , ,

1

2, , , , , , ,

1

2, , , , , , ,

1

, ,, , ,

1

, , ,

1 [ ( 2 )6

1 [( ) 2 ]6

1 ( 2 )6

1 ( 2 )6

12

k

, ]x j x j x j x j x j x j x x jj

k

x j x j x x j x j x j x j x jj

k

x j x j x j x j x j x j x jj

kx j x j

x j x j x jj j j

x j x j x j

D A h u h u u u

A h u A h u u

A h u A h u u

u uA A h

n nu

A A h

ψ

ψ

=

=

=

=

= − Δ + Δ − Δ

= − Δ − +

= − Δ +

= − Δ +

= − Δ

,

1

1 ,2

kx j

j j

IIx

n

S

=

= −

(5-8d)

here is the Seidel coma for x-RSOS. IIxS

Page 92: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

91

For , we find 5D

2 25 , , ,

1

,, , ,

1

,, , ,

1

1 ( )2

1 ( )( )(2

1 .2

k

y j y j x j y x jj

k

,

)x jy j y j y j x j

j j

kx j

x j y j y jj j

D A h u u

uA h n u

nu

A A hn

ψ

ψ

=

=

=

= − Δ − Δ

= − Δ −

= − Δ

(5-8e)

For , we find 6D

6 , , ,

1

,, , ,

1

12

1 .2

k

y j y j x j x jj

k

,

x jx j y j y j

j j

D A h u u

uA A h

n

=

=

= − Δ

= − Δ

∑ (5-8f)

For , we find 7D

2 27 , , , , , , , , ,

1

2, , , , ,

2, , , , , , , , , , ,

1

2, , ,

, , , , ,

1 [ ( 26

2 ) ]

1 [ ( 2 2 )6

( ) ]

1 [2 (6

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j

k

y j y j y j y j y j y j y j y j y j y j y jj

y j y j y y j

y j y j y j y j y j y

D A h u c h u h u u

c h h u u

A c h u h u u c h h u

A h u

A h u u c h

ψ

ψ

=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ + +

+ −

= − Δ +

,2 2, , , , , , , , ,

1

, , ,2, , , , , , , , , , ,

1

, ,2 2, , , , , , , , , ,

) ( )

1 [2 ( )]6

1 (3 )6

ky j ]j y j y j y j y j y j y j y j y j

j j

ky j y j y j

y j y j y j y j y j y j y j y j y j y j y jj j j j

y j y jy j y j y j y j y j y j y j y j y j y j

j j j

AA h c h c A h u

n

u c AA A h A A h A h u h c

n n n

u AA A h A h c A h c u

n n

=

=

+ − +

= − Δ + + −

= − Δ + −

1

, ,2, , , , , , ,

1

, 3 2, , , , , ,

1

1 [3 ( )]6

1 (3 )6

k

ky j y j

y j y j y j y j y j y j y jj j j

ky j

y j y j y j y j y j y jj j

u AA A h A h c u

n n

uA A h A h c

n

=

=

=

= − Δ + −

= − Δ +

Page 93: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

92

,, , ,

1

121 ,2

ky j

y j y j y jj j

IIy

uA A h

n

S

=

= − Δ

= −

∑ (5-8g)

here IIyS is the Seidel coma for y-RSOS. In the above deduction, we have made use of the

relationship 3 2, , ,( )y j y j y jA h cΔ = 0 since 3 2

, , ,y j y j y jA h c is a refraction constant. We will no

longer mention this kind of detail in later deduction procedures.

For , we find 8D

28 , , , , , ,

1

2, , , , , , ,

1

, ,2 2, , ,

1

,2, ,

1

1 [ ( 2 ) 24

1 ( 2 )4

1 [ ( 2 )]4

343 ,4

k

, , ]x j x j x j x j x j x j x x j x jj

k

x j x j x j x j x j x j x jj

kx j x j

x j x j x jj j j

kx j

x j x jj j

IIIx

D A h u h u u u

A h u A h u u

u uh A A

n nu

A hn

S

ψ=

=

=

=

= − Δ + Δ − Δ

= − Δ +

= − Δ +

= − Δ

= −

u

(5-8h)

here is the Seidel astigmatism for x-RSOS. IIIxS

For , we find 9D

2 29 , , , , , , , , , , , , ,

1

, ,

2 2, , , , , , , , , , , , , ,

1

2 2, , , , ,

1 [ ( 2 2 )4

2 ]

1 [ ( 2 ) 2 ]4

1 [ (4

k

y j y j y j y j y j y j y j y j y j y j y j y j y jj

y y j y j

k

y j y j y j y j y j y j y j y j y j y j y j y j y j y jj

y j y j y j y j y j y

D A h u c h u h u u c h h u

u u

A h u c h u c h h u A h u u

A h u c h u

ψ=

=

= − Δ + Δ + Δ + Δ

− Δ

= − Δ + + +

= − Δ +

,, , , , , , , , , ,

1

2 ) 2 (k

y j )]j y j y j y j y j y j y j y j y j y jj j

Ac h h u A h u h c

n=

+ + −∑

Page 94: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

93

, ,2 2 2, , , , , , , , , , , , ,

1

,, , , , , ,

, ,2 2, , , , , ,

1

,, , ,

1 {2 [ 2 ( )]4

2 ( )}

1 {2 [ ( )4

2 (

ky j y j

y j y j y j y j y j y j y j y j y j y j y j y j y jj j j

y jy j y j y j y j y j y j

j

ky j y j

y j y j y j y j y j y jj j j

y jy j y j y j

j

u AA h A h u c h h u c h h c

n nA

A h h c h cn

u AA h A h h c

n n

Ac h h

n

=

=

= − Δ + + + −

− −

= − Δ + −

+

, ,2 2, , , , , , , ,

, ,2 2 2 2, , , , , , , , , , ,2

1

, , , ,2 2 2 2, , , , , ,

)] 2 }

1 1(2 2 )4

( )1 [24

y j y jy j y j y j y j y j y j y j y j

j j

ky j y j y j

y j y j y j y j y j y j y j y j y j y j y jj j j j

y j y j y j y jy j y j y j y j y j y j

j j

c ch c A h A A h h

n nu c

A h A A h A h A A h hn n n

u u h cA h A h A h

n n

=

− + −

= − Δ + + −

+= − Δ + +

∑ ,

j

cn

,

1

,, , , ,

, ,2 2 2 2 2, , , , , , , , , ,

1

,2 2, , ,

1

2 ]

1 [3 ( 2 ) ]4

1 (3 )41 (3 ),4

ky j

j j

y jy j y j y j y j

j

ky j y j

y j y j y j y j y j y j y j y j y j y jj j j

ky j

y j y j y y jj j

IIIy IVy

cn

cA A h h

nu c

A h A h A h A A h hn n

uA h P

n

S S

=

=

=

= − Δ + + −

= − Δ + Ψ

= − +

∑ (5-8i)

here IIIyS and IVyS are the Seidel astigmatism and Petzval sums for y-RSOS. The

combination of both terms is often called field curvature.

For and , we will keep the terms unchanged as 10D 11D

2 210 , , , , , ,

1

1 (4

k

).x j x j y j y j y j x jj

D A h u c h=

= − Δ + Δ∑ u (5-8j)

211 , , ,

1

1 .4

k

y j y j x jj

D A h=

= − Δ∑ u (5-8k)

For , we find 12D

Page 95: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

94

12 , , , , , ,

1

,, , ,

1

( )

.

k

y j y j x j x j y x j x jj

kx j

x j y j y jj j

D A h u u u

uA A h

n

ψ=

=

= − Δ − Δ

= − Δ

u (5-8 l )

For , we find 13D

2 213 , , , ,

1

2, , ,

1

3, , 2

1

1 ( )2

12

1 12

1 ,2

k

x j x j x j x x jj

k

x j x j x jj

k

x j x jj j

Vx

D A h u

A h u

A hn

S

ψ=

=

=

= − Δ − Δ

= − Δ

= − Δ

= −

u

(5-8m)

here is the Seidel distortion for x-RSOS. VxS

For , we find 14D

2 2 214 , , , , , , ,

1

2 2, , , , , , ,

1

, ,2 2, , , , , , , , ,

1

3 2, , , , ,

2

1 [ ( ) ]2

1 ( )2

1 [ ( ) (2

21 (2

k

y j y j y j y j y j y j y y jj

k

y j y j y j y j y j y j y jj

ky j y j

y j y j y j y j y j y j y j y j y jj j j

y j y j y j y j y j

j

D A h u c h u u

A c h u A h u

A AA c h h c A h h c

n n

A h A h h cn

ψ=

=

=

= − Δ + Δ − Δ

= − Δ +

= − Δ − + −

= − Δ −

∑ ) ]

, ,2, , ,

1

, ,3 2 2, , , , , , , ,2

1

3, , , , , , , , ,2

1

3, , , , , ,2

)

1 1( 22

1 1[ ( 2 )2

1 1[ ( )2

ky j y j

y j y j y jj j j

ky j y j

y j y j y j y j y j y j y j y jj j j

k

y j y j y j y j y j y j y j y j y jj j

y j y j y j y j y y j y j yj

cA A h

n nc c

A h A h h A A hn n

A h A h A h A h Pn

A h A h A h Pn

=

=

=

+

= − Δ − Δ + Δ

= − Δ + −

= − Δ + Ψ −

)

]

jn

,1

]

1 ,2

k

jj

VyS

=

= −

(5-8n)

Page 96: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

95

here is the Seidel distortion for y-RSOS [28]. VyS

For , we find 15D

2 215 , , , , , , ,

1

2 2, , , , , , ,

1

1 [ ( )2

1 ( )2

k2 ]

.

x j x j y j y j y j x j x y jj

k

x j x j y j x j y j y j x jj

D A h u c h u

A h u A c h u

ψ=

=

= − Δ + Δ − Δ

= − Δ + Δ

u (5-8o)

For , we find 16D

2 216 , , , ,

1

2, , ,

1

1 ( )2

1 .2

k

y j y j x j y x jj

k

y j y j x jj

D A h u

A h u

ψ=

=

= − Δ − Δ

= − Δ

u (5-8p)

5.5 Summary

In this chapter, we showed the primary wave aberration coefficients for parallel

cylindrical anamorphic attachment systems in subgroups as:

Primary wave aberration coefficients associated with x-RSOS

,21 , ,

1

1 18 8

kx j

x j x j Ixj j

uD A h

n=

= − Δ = −∑ S ,

,4 , , ,

1

1 12 2

kx j

x j x j x j IIxj j

uD A A h

n=

= − Δ = −∑ S ,

,28 , ,

1

3 34 4

kx j

x j x j IIIxj j

uD A h

n=

= − Δ = −∑ S ,

313 , , 2

1

1 12 2

k 1x j x j Vx

j j

D A hn=

= − Δ = −∑ S .

Page 97: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

96

In the above expression etc. mean the results are in the same form as the Seidel

aberration terms corresponding to the rays trace in the x-z cross section of the associated

x-RSOS, which are parallel planes in the current case. Thus for rays staying in the x-z

symmetry plane of the parallel cylindrical anamorphic attachment system, their

aberration behavior is the same as rays staying in the x-z meridian section of the

associated x-RSOS.

,Ix IIxS S

Primary wave aberration coefficients associated with y-RSOS

,22 , ,

1

1 18 8

ky j

y j y j Iyj j

uD A h

n=

= − Δ = −∑ S ,

,7 , , ,

1

1 12 2

ky j

y j y j y j IIyj j

uD A A h

n=

= − Δ = −∑ S ,

,2 29 , , ,

1

1 1(3 ) (3 )4 4

ky j

y j y j y y j IIIy IVyj j

uD A h P S

n=

= − Δ + Ψ = − +∑ S ,

314 , , , , , , ,2

1

1 1[ ( )2 2

k

y j y j y j y j y y j y j y j Vyj j

D A h A h A h Pn=

= − Δ + Ψ − = −∑ 1] S .

Similarly, in the above expression ,Iy IIyS S etc. mean the results are in the same form as

the Seidel aberration terms corresponding to the rays trace in the y-z cross section of the

associated y-RSOS. Thus for rays staying in the y-z symmetry plane of the parallel

cylindrical anamorphic attachment system, their aberration behavior is the same as rays

staying in the y-z meridian section of the associated y-RSOS. Actually, these conclusions

are generally applicable for any types of anamorphic systems because all rays in the two

symmetry planes are non-skewed.

Additional terms for skew rays

Page 98: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

97

,3 , , ,

1

14

kx j

x j y j y jj j

uD A A h

n=

= − Δ∑ ,

,5 , , ,

1

12

kx j

x j y j y jj j

uD A A h

n=

= − Δ∑ ,

,6 , , ,

1

12

kx j

x j y j y jj j

uD A A h

n=

= − Δ∑ ,

2 210 , , , , , ,

1

1 ( )4

k

x j x j y j y j y j x jj

D A h u c h=

= − Δ + Δ∑ u ,

211 , , ,

1

14

k

y j y j x jj

D A h=

= − Δ∑ u ,

,12 , , ,

1

kx j

x j y j y jj j

uD A A h

n=

= − Δ∑ ,

2 215 , , , , , , ,

1

1 ( )2

k

x j x j y j x j y j y j x jj

D A h u A c h=

= − Δ + Δ∑ u ,

216 , , ,

1

12

k

y j y j x jj

D A h=

= − Δ∑ u .

These expressions tell us that for skew rays not staying in one of the symmetry planes of

the parallel cylindrical anamorphic system, their aberration behavior will be much more

complex than in the RSOS case and will possess of all 16 anamorphic primary aberration

types.

We emphasize that all parameters in the anamorphic primary aberration coefficient

expressions are the paraxial marginal and chief rays’ tracing data in the two associated

RSOS, together with some first-order constants and definitions. Thus, for anamorphic

Page 99: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

98

primary aberration calculation purpose, we only need to trace the four non-skew marginal

and chief rays, in the associated x-RSOS and y-RSOS, respectively.

If we compare the above expressions with what C. G. Wynne reports in his 1954

paper [13], we will see the results are the same, but our development did not put any

restriction on location of the system stop, thus is more general than Wynne’s treatment.

Notice that parallel cylindrical anamorphic attachment systems themselves are not

imaging systems due to the fact that all elements have no power in one principal section.

To form an anamorphic image, we can use the above cylindrical elements as an

anamorphic attachment and combine it with some ordinary imaging lenses to obtain

optical power in the natural plane, as shown in Figure 5-1 above.

We also notice that the parallel cylindrical anamorphic attachment have an

interesting property. As Wynne noted in his 1954 paper, for an object at infinity, because

, ,x j j x jA n u= will be zero for all surfaces in the attachment, aberration types

1 3, 4 5 6 8 1, , , , , 0D D D D D D D and will disappear automatically. The remaining aberrations

term are the four y-RSOS aberrations

12D

2 7, 9 14, ,D D D D , together with three distortion terms

13 15, 16,D D D and one astigmatism term . 11D

However, if the object distance is finite ( 0xu ≠ ), all sixteen primary aberration types

will survive.

Page 100: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

99

CHAPTER 6

PRIMARY ABERRATION THEORY FOR CROSS CYLINDRICAL

ANAMORPHIC SYSTEMS

In chapter 5, the 16 primary wave aberration coefficients for parallel cylindrical

anamorphic systems were obtained. In this chapter, we will apply the same method onto

cross cylindrical anamorphic systems.

Following this same development pattern as described in chapter 5, we will present

primary ray aberration coefficients for cross cylindrical anamorphic systems in section

6.1, primary wave aberration coefficients for cross cylindrical anamorphic systems and

their simplification in sections 6.2 and 6.3.

6.1 Primary ray aberration coefficients for cross cylindrical anamorphic systems

For cylindrical anamorphic systems, the next configurations to consider is a system

made from cross cylindrical lenses, as shown in the Figure 6-1 below.

Figure 6-1 A cross cylindrical anamorphic system example

Page 101: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

100

The two principal sections of this kind of system are shown in Figure 6-2 below.

Rays propagating in one of the principal sections will pass through the cylindrical system

as if the system were made from a mixture of spherical surfaces and plane surfaces.

Figure 6-2 The two principal sections of a cross cylindrical anamorphic system

Notice that a plane surface is a special case of a spherical surface with radius r = ∞ ,

thus we can write the surface equation for any surface j up to the second order as

2 22 , 1 , 1

1 ( )2j x j j y j jz c x c y= + , (6-1)

of which either ,x jc or ,y jc will be zero. Thus any term containing , ,x j yc c× j will be zero.

From Chapter 4.4, we know

1 ,j x x j x x , jx h H hρ= + ,

1 ,j y y j y yy h H hρ= + , j ,

1 , ,j x x j x x jL u H uρ= + ,

1 , ,j y y j y y jM u H uρ= + ,

2 2

1 1 22 , , ,

1 [( ) ( ) ]2 2

j jj x x j x x j y y j

L MN u H u uδ ρ ρ

+= = + + + 2

,y y jH u .

Page 102: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

101

By putting equation (6-1) together with the above equations into equations (5-3), we get:

' ' ', 3 , 1 2 2 1 2

1 1

2 2 2 2, , , , , , ,

1

2 2 2 2, , , , , , ,

1

,1

[ ( ) ]

1 [( )( 2 )2

1 [( )( 2 )2

1 [2

k k

k x k k x j j j j j x x jj j

k

x j x x j x x j x x j x x j x x x j x jj

k

x j x x j x x j y y j y y j y y y j y jj

k

x jj

n u A x N z L H N

A h H h u H u H u u

A h H h u H u H u u

A c

δξ δ ψ δ

ρ ρ ρ

ρ ρ ρ

= =

=

=

=

= Δ − + +

= − + Δ + Δ + Δ

− + Δ + Δ + Δ

∑ ∑

∑ 2 2 2 2, , , , , , ,

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , ,

1

2 2 2 2, ,

( 2 )( )]

1 [ ( 2 )( )2

1 ( 2 )2

1 ( 22

x j x j x x j x j x x x j x x x j x x j

k

]x j y j y j y y j y j y y y j y x x j x x jj

k

x x x x j x x j x x x j x jj

x x y y j y y j

h h h H h H u H u

A c h h h H h H u H u

H u H u H u u

H u H u

ρ ρ ρ

ρ ρ ρ

ψ ρ ρ

ψ ρ ρ

=

=

+ + Δ + Δ

− + + Δ +

+ Δ + Δ + Δ

+ Δ + Δ +

Δ

, ,1

),k

y y y j y jj

H u u=

Δ∑

(6-2a)

And,

' ' ', 3 , 1 2 2 1 2

1 1

2 2 2 2, , , , , , ,

1

2 2 2 2, , , , , , ,

1

,1

[ ( ) ]

1 [( )( 2 )2

1 [( )( 2 )2

1 [2

k k

k y k k y j j j j j y y jj j

k

y j y y j y y j x x j x x j x x x j x jj

k

y j y y j y y j y y j y y j y y y j y jj

k

y jj

n u A y N z M H N

A h H h u H u H u u

A h H h u H u H u u

A c

δη δ ψ δ

ρ ρ ρ

ρ ρ ρ

= =

=

=

=

= Δ − + +

= − + Δ + Δ + Δ

− + Δ + Δ + Δ

∑ ∑

∑ 2 2 2 2, , , , , , ,

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , ,

1

2 2 2 2, ,

( 2 )(

1 [ ( 2 )( )2

1 ( 2 )2

1 ( 22

x j x j x x j x j x x x j x y y j y y j

k

y j y j y j y y j y j y y y j y y y j y y jj

k

y y x x j x x j x x x j x jj

y y y y j y y j

h h h H h H u H u

A c h h h H h H u H u

H u H u H u u

H u H u

ρ ρ ρ

ρ ρ ρ

ψ ρ ρ

ψ ρ ρ

=

=

+ + Δ + Δ

− + + Δ +

+ Δ + Δ + Δ

+ Δ + Δ +

)]

, ,1

).k

y y y j y jj

H u u=

Δ∑

(6-2b)

Page 103: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

102

By regrouping the above equations by their field and aperture dependences, we get

the corresponding primary ray aberrations terms in a form similar to equations (3-8) as:

For Spherical Aberration-like aberration types

1 :D' ' ' 2 2 3

, 3 , , , , , ,1

' ' ', 3

1 [ ( )]2

0

k

k x k k x j x j x j x j x j x j xj

k y k k

n u A h u c h u

n u

δξ ρ

δη=

⎫= − Δ + Δ ⎪

⎬⎪= ⎭

∑ (6-3a)

2 :D

' ' ', 3

' ' ' 2 2 3, 3 , , , , , ,

1

0

1 [ ( )]2

k x k k

k

k y k k y j y j y j y j y j y j yj

n u

n u A h u c h u

δξ

δη ρ=

⎫=⎪⎬

= − Δ + Δ ⎪⎭

∑ (6-3b)

3 :D

' ' ' 2 2 2, , , , , , ,

1

' ' ' 2 2 2, , , , , ,

1

1 [ ( )]2

1 [ ( )]2

k

k x k k x j x j y j y j y j x j x yj

k

k y k k y j y j x j x x j y j x yj

n u A h u c h u

n u A h u c h u

δξ ρ ρ

δη ρ ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎬⎪= − Δ + Δ⎪⎭

∑ (6-3c)

For Coma-like aberration types

4 :D

' ' ' 2 2, 3 , , , , , ,

1

2, , , , , , , ,

' ' ', 3

1 [ (2

2 2 ) ]

0

k

k x k k x j x j x j x j x j x jj

x j x j x j x j x j x j x j x x j x x

k y k k

n u A h u c h u

h u u c h h u u H

n u

δξ

ψ

δη

=

⎫= − Δ + Δ ⎪

⎪⎪+ Δ + Δ − Δ ⎬⎪

= ⎪⎪⎭

∑2ρ (6-3d)

5 :D

' ' ', 3 , , , , , , , ,

1

' ' ' 2 2 2 2, 3 , , , , , , ,

1

[ ( )]

1 [ ( ) ]2

k

k x k k x j x j y j y j y j y j y j x j y x yj

k

k y k k y j y j x j x j x j y j y x j y xj

n u A h u u c h h u H

n u A h u c h u u H

δξ ρ ρ

δη ψ

=

=

⎫= − Δ + Δ

ρ

⎪⎪⎬⎪= − Δ + Δ − Δ⎪⎭

∑ (6-3e)

Page 104: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

103

6 :D

' ' ' 2 2 2 2, 3 , , , , , , ,

1

' ' ', 3 , , , , , , , ,

1

1 [ ( ) ]2

[ ( )]

k

k x k k x j x j y j y j y j x j x y j x yj

k

k y k k y j y j x j x j x j x j x j y j x x yj

n u A h u c h u u H

n u A h u u c h h u H

δξ ρ

δη ρ ρ

=

=

⎫= − Δ + Δ −Ψ Δ ⎪

⎪⎬⎪= − Δ + Δ⎪⎭

∑ (6-3f)

7 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

2 2, , , , , , , ,

0

1 [ (2

2 2 ) ]

k x k k

k

k y k k y j y j y j y j y j y jj

y j y j y j y j y j y j y j y y j y y

n u

n u A h u c h u

h u u c h h u u H

δξ

δη

ψ ρ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ Δ + Δ − Δ ⎭

∑ (6-3g)

For Astigmatism and Field curvature-like aberration types

8 :D

' ' ' 2 2, 3 , , , , , ,

1

2, , , , , , , , ,

' ' ', 3

1 [ (2

2 2 ) 2 ]

0

k

k x k k x j x j x j x j x j x jj

x j x j x j x j x j x j x j x x j x j x x

k y k k

n u A h u c h u

h u u c h h u u u H

n u

δξ

ψ ρ

δη

=

⎫= − Δ + Δ ⎪

⎪⎪+ Δ + Δ − Δ ⎬⎪

= ⎪⎪⎭

∑ (6-3h)

9 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

2, , , , , , , , ,

0

1 [ (2

2 2 ) 2 ]

k x k k

k

k y k k y j y j y j y j y j y jj

y j y j y j y j y j y j y j y y j y j y y

n u

n u A h u c h u

h u u c h h u u u H

δξ

δη

ψ ρ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ Δ + Δ − Δ ⎭

∑ (6-3i)

10 :D' ' ' 2 2 2

, 3 , , , , , ,1

' ' ', 3

1 [ ( )]2

0

k

k x k k x j x j y j y j y j x j y xj

k y k k

n u A h u c h u H

n u

δξ ρ

δη=

⎫= − Δ + Δ ⎪

⎬⎪= ⎭

∑ (6-3j)

11 :D

' ' ', 3

' ' ' 2 2 2, 3 , , , , , ,

1

0

1 [ ( )]2

k x k k

k

k y k k y j y j x j x j x j y j x yj

n u

n u A h u c h u H

δξ

δη ρ=

⎫=⎪⎬

= − Δ + Δ ⎪⎭

∑ (6-3k)

Page 105: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

104

12 :D

' ' ', 3 , , , , , , , ,

1

, ,

' ' ', 3 , , , , , , , ,

1

, ,

[ ( )

]

[ ( )

]

k

k x k k x j x j y j y j y j y j y j x jj

x y j y j x y y

k

k y k k y j y j x j x j x j x j x j y jj

y x j x j x y x

n u A h u u c h h u

u u H H

n u A h u u c h h u

u u H H

δξ

ψ ρ

δη

ψ ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎪− Δ ⎪⎬⎪= − Δ + Δ⎪⎪

− Δ ⎪⎭

∑ (6-3l)

For Distortion-like aberration types

13 :D' ' ' 2 2 2 3

, 3 , , , , , , ,1

' ' ', 3

1 [ ( ) ]2

0

k

k x k k x j x j x j x j x j x j x x j xj

k y k k

n u A h u c h u u H

n u

δξ ψ

δη=

⎫= − Δ + Δ − Δ ⎪

⎬⎪= ⎭

∑ (6-3m)

14 :D

' ' ', 3

' ' ' 2 2 2 3, 3 , , , , , , ,

1

0

1 [ ( ) ]2

k x k k

k

k y k k y j y j y j y j y j y j y y j yj

n u

n u A h u c h u u H

δξ

δη ψ=

⎫=⎪⎬

= − Δ + Δ − Δ ⎪⎭

∑ (6-3n)

15 :D' ' ' 2 2 2 2

, 3 , , , , , , ,1

' ' ', 3

1 [ ( ) ]2

0

k

k x k k x j x j y j y j y j x j x y j x yj

k y k k

n u A h u c h u u H H

n u

δξ ψ

δη=

⎫= − Δ + Δ − Δ ⎪

⎬⎪= ⎭

∑ (6-3o)

16 :D

' ' ', 3

' ' ' 2 2 2 2, 3 , , , , , , ,

1

0

1 [ ( ) ]2

k x k k

k

k y k k y j y j x j x j x j y j y x j x yj

n u

n u A h u c h u u H H

δξ

δη ψ=

⎫=⎪⎬

= − Δ + Δ − Δ ⎪⎭

∑ (6-3p)

6.2 Primary wave aberration coefficients for cross cylindrical anamorphic systems

By comparing equations (6-3) with equations (3-8), we immediately get the primary

wave aberration coefficients though as 1D 16D

2 21 , , , , ,

1

1 (8

k

, )x j x j x j x j x j x jj

D A h u c h=

= − Δ + Δ∑ u (6-4a)

Page 106: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

105

2 22 , , , , ,

1

1 (8

k

y j y j y j y j y j y jj

D A h u c h=

= − Δ + Δ∑ , )u (6-4b)

2 23 , , , , ,

1

1 (4

k

y j y j x j x j x j y jj

D A h u c h=

= − Δ + Δ∑ , )u (6-4c)

2 2

4 , , , , , , ,1

2, , , , ,

1 [ ( 26

2 ) ]

k

, ,x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j

D A h u c h u h u

c h h u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ u (6-4d)

2 25 , , , , , ,

1

1 [ ( )2

k

y j y j x j x j x j y j y x jj

D A h u c h u uψ=

= − Δ + Δ − Δ∑ 2, ] (6-4e)

6 , , , , , , , ,1

1 (2

k

y j y j x j x j x j x j x j y jj

D A h u u c h h=

= − Δ + Δ∑ )u (6-4f)

2 2

7 , , , , , , ,1

2, , , , ,

1 [ ( 26

2 ) ]

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j

D A h u c h u h u

c h h u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ , ,u (6-4g)

2 2

8 , , , , , , ,1

, , , , , ,

1 [ ( 24

2 ) 2 ]

k

, ,x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j x j

D A h u c h u h u

c h h u u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ u (6-4h)

2 2

9 , , , , , , ,1

, , , , , ,

1 [ ( 24

2 ) 2 ]

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j

D A h u c h u h u

c h h u u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ , ,u (6-4i)

2 210 , , , , , ,

1

1 (4

k

)x j x j y j y j y j x jj

D A h u c h=

= − Δ + Δ∑ u (6-4j)

2 211 , , , , , ,

1

1 (4

k

y j y j x j x j x j y jj

D A h u c h=

= − Δ + Δ∑ )u (6-4k)

12 , , , , , , , , , ,1[ ( )

k

y j y j x j x j x j x j x j y j y x j x jj

D A h u u c h h u uψ=

= − Δ + Δ − Δ∑ ]u (6-4l)

Page 107: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

106

2 213 , , , , , , ,

1

1 [ ( )2

k2 ]x j x j x j x j x j x j x x j

jD A h u c h u ψ

=

= − Δ + Δ − Δ∑ u (6-4m)

2 214 , , , , , , ,

1

1 [ ( )2

k

y j y j y j y j y j y j y y jj

D A h u c h u ψ=

= − Δ + Δ − Δ∑ 2 ]u (6-4n)

2 215 , , , , , , ,

1

1 [ ( )2

k2 ]x j x j y j y j y j x j x y j

jD A h u c h u ψ

=

= − Δ + Δ − Δ∑ u (6-4o)

2 216 , , , , , , ,

1

1 [ ( )2

k

y j y j x j x j x j y j y x jj

D A h u c h u ψ=

= − Δ + Δ − Δ∑ 2 ]u (6-4p)

Again, for , we can either choose the coefficient in the x-ray error

expression or the coefficient in the y-ray error expression. The numerical values may

differ for any single surface

3 5 6 12, , ,D D D D

j , but the summation in the final image space will be the

same. In our treatment above, the coefficient in the y-ray error expression was arbitrarily

chosen.

6.3 Simplification of the results

Now we have all the primary aberration coefficients for a cross cylindrical

anamorphic system in terms of the paraxial marginal and chief ray trace data in both

associated RSOS. Similar to the treatment in chapter 5, let us use the corresponding

paraxial definitions from chapter 2.11 to simplify the results so that we can rewrite

equations (6-4) in a form as similar to the Seidel aberrations for RSOS as possible. For

the current configuration, we have

, , , ,x j j x j j x j x jA n u n h c= + ,

Page 108: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

107

, , , ,x j j x j j x j x jA n u n h c= + ,

, , , ,y j j y j j y j y jA n u n h c= + ,

, , ,y j j y j j y j y j,A n u n h c= + .

Again, notice that either ,x jc or ,y jc will be zero for surface j , thus any term containing

, ,x j yc c× j will be zero. Other paraxial definitions are the same of chapter 5.

For , we find 1D

2 21 , , , , ,

1

, , , , , ,1

,2, ,

1

1 ( )8

1 ( )8

181 ,8

k

,x j x j x j x j x j x jj

k

x j x j x j x j x j x jj

kx j

x j x jj j

Ix

D A h u c h

A h u u h c

uA h

n

S

=

=

=

= − Δ + Δ

= − Δ +

= − Δ

= −

u

(6-5a)

here and below ,IxS IyS , etc. will have the same meaning as in chapter 5.

For , we find 2D

2 22 , , , , ,

1

, , , , , ,1

,2, ,

1

1 ( )8

1 ( )8

18

1 .8

k

y j y j y j y j x j y jj

k

y j y j y j y j y j y jj

ky j

y j y jj j

Iy

D A h u c h

A h u u c h

uA h

n

S

=

=

=

= − Δ + Δ

= − Δ +

= − Δ

= −

,u

(6-5b)

For , we will keep it unchanged as 3D

2 23 , , , , ,

1

1 (4

k

y j y j x j x j x j y jj

D A h u h c u=

= − Δ + Δ∑ , ). (6-5c)

Page 109: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

108

For , we find 4D

2 24 , , , , , , , , ,

1

2, , , , ,

2 2, , , , , , , , , , , , , ,

1

2 2, , , , ,

1

1 [ ( 26

2 ) ]

1 [ ( 2 26

1 [6

k

x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j

k

)]x j x j x j x j x j x j x j x j x j x j x j x j x j x jj

k

x j x j x j x j x j xj

D A h u c h u h u u

c h h u u

A h u A c h u c h h u h u u

A h u A c h

ψ=

=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ + + +

= − Δ +

∑ , , , , , , , ,

, ,2, , , , , , , , , ,

1

,, , ,

, 2 2, , , , , , , , , , ,

1

2 ( )

1 [ ( ) ( )6

2 ]

1 1(3 )6

j x j x j x j x j x j x j x j

kx j x j

x j x j x j x j x j x j x j x j x j x jj j j

x jx j x j x j

j

kx j

x j x j x j x j x j x j x j x j x j x j x jj j j

u A h u u c h

A AA h u c h A c h c h

n nu

A A hn

uA A h A A c h A h c u

n n

=

=

+ +

= − Δ − + −

+

= − Δ + −

= −

]

, ,2, , , , , , ,

1

, 2 3, , , , , ,

1

,, , ,

1

1 [3 ( )]6

1 (3 )6

121 .2

kx j x j

x j x j x j x j x j x j x jj j j

kx j

x j x j x j x j x j x jj j

kx j

x j x j x jj j

IIx

u AA A h A c h u

n nu

A A h A c hn

uA A h

n

S

=

=

=

Δ + −

= − Δ +

= − Δ

= −

∑ (6-5d)

For , we find 5D

2 2 25 , , , , , ,

1

2 2, , , , , , ,

1

1 [ ( )2

1 ( )2

k

y j y j x j x j x j y j y x jj

k

y j y j x j y j x j x j y jj

D A h u c h u u

A h u A h c u

ψ=

=

= − Δ + Δ − Δ

= − Δ + Δ

, ]

. (6-5e)

For , we will keep it unchanged as 6D

6 , , , , , , ,1

1 ( )2

k

y j y j x j x j x j x j x j y jj

D A h u u h h c u=

= − Δ + Δ∑ , . (6-5f)

Page 110: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

109

For , by noticing that equation (6-4g) is exactly the same as equation (5-7g), we

know

7D

2 27 , , , , , , ,

1

2, , , , ,

,, , ,

1

1 [ ( 26

2 ) ]

121 .2

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j

ky j

y j y j y jj j

IIy

D A h u c h u h u

c h h u uu

A A hn

S

ψ=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ

= −

, ,u

(6-5g)

For , we find 8D

2 28 , , , , , , , , ,

1

, , , , , ,

2 2, , , , , , , , , , , , , ,

1

,, , , ,

1 [ ( 24

2 ) 2 ]

1 [ ( 2 ) 2 ]4

1 { [ ( )4

k

x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j x j

k

x j x j x j x j x j x j x j x j x j x j x j x j x j x jj

x jx j x j x j x j

j

D A h u c h u h u u

c h h u u u

A h u c h u c h h u A h u u

AA h h c

n

ψ=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ + + +

= − Δ −

,2 2, , , ,

1

, , ,, , , , , , , , , , ,

,2 2 2 2 2, , , , , , , , , , ,2

1

,

( )

2 ( )] 2 ( )( )}

1 1[3 ( 2 2 )]4

1 {34

kx j

x j x j x j x jj j

x j x j x jx j x j x j x j x j x j x j x j x j x j x j

j j j

kx j

x j x j x j x j x j x j x j x j x j x j x jj j j

x j

Ac h h c

n

A A Ac h h h c A h h c h c

n n n

cA A h A h A h A A h h

n n

A

=

=

+ −

+ − + − −

= − Δ + − −

= − Δ

,2 2, , , , , , , ,2

1

, ,2 2 2, , , , , ,2

1

,2 2, , ,

1

1 [( ) 3 ]}

1 1 1[3 3 ( )]4

1 (3 )4

1 (3 ).4

kx j

x j x j x j x j x j x j x j x jj j j

kx j x j

x j x j x j x x j x j x jj j j j j

kx j

x j x j x x jj j

IIIx IVx

cA h A h A h A h

n n

c AA A h A h u

n n n n

uA h P

n

S S

=

=

=

+ − −

= − Δ + Ψ − −

= − Δ + Ψ

= − +

2 2

(6-5h)

Page 111: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

110

For , by noticing that equation (6-4i) is exactly the same as equation (5-7i), we

know:

9D

2 29 , , , , , , ,

1

, , , , , ,

,2 2, , ,

1

1 [ ( 24

2 ) 2 ]

1 (3 )41 (3 ).4

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j

ky j

y j y j y y jj j

IIIy IVy

D A h u c h u h u

c h h u u uu

A h Pn

S S

ψ=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ + Ψ

= − +

, ,u

(6-5i)

For and , we will keep them unchanged as 10D 11D

2 210 , , , , , ,

1

1 (4

k

).x j x j y j y j y j x jj

D A h u h c=

= − Δ + Δ∑ u (6-5j)

2 211 , , , , , ,

1

1 (4

k

y j y j x j x j x j y jj

D A h u h c u=

= − Δ + Δ∑ ). (6-5k)

For , we find 12D

12 , , , , , , , , , ,

1

, , , , , , , , ,1

[ ( )

( )

k

y j y j x j x j x j x j x j y j y x j x jj

k

y j y j x j x j y j x j x j x j y jj

D A h u u c h h u u

A h u u A h h c u

ψ=

=

= − Δ + Δ − Δ

= − Δ + Δ

]

.

u (6-5 l )

For , we find 13D

2 2 213 , , , , , , ,

1

2 2, , , , , , ,

1

, ,2 2, , , , , , , , ,

1

3 2, , , , ,2

1 [ ( ) ]2

1 ( )2

1 [ ( ) (2

1 1( 22

k

x j x j x j x j x j x j x x jj

k

x j x j x j x j x j x j x jj

kx j x j )]x j x j x j x j x j x j x j x j x j

j j j

x j x j x j x j x jj

D A h u c h u u

A h u A c h u

A AA h h c A c h h c

n n

A h A h hn

ψ=

=

=

= − Δ + Δ − Δ

= − Δ +

= − Δ − + −

= − Δ −

, ,2, , ,

1

)k

x j x jx j x j x j

j j j

c cA A h

n n=

+∑

(6-5m)

Page 112: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

111

3, , , , , , , , ,2

1

3, , , , , , ,2

1

1 1[ ( 22

1 1[ ( )2

1 .2

k

) ]

]

x j x j x j x j x j x j x j x j x jj j

k

x j x j x j x j x x j x j x jj j

Vx

A h A h A h A h Pn

A h A h A h Pn

S

=

=

= − Δ + −

= − Δ + Ψ −

= −

For , by noticing that equation (6-4n) is exactly the same as equation (5-7n), we

know

14D

2 2 214 , , , , , , ,

1

3, , , , , , ,2

1

1 [ ( ) ]2

1 1[ (21 .2

k

y j y j y j y j y j y j y y jj

k

y j y j y j y j y y j y j y jj j

Vy

D A h u c h u

A h A h A h Pn

S

ψ=

=

= − Δ + Δ − Δ

= − Δ + Ψ −

= −

∑ ) ]

u

(6-5n)

For , we find 15D

2 215 , , , , , , ,

1

2 2, , , , , , ,

1

1 [ ( )2

1 ( )2

k2 ]

.

x j x j y j y j y j x j x y jj

k

x j x j y j x j y j y j x jj

D A h u c h u

A h u A h c u

ψ=

=

= − Δ + Δ − Δ

= − Δ + Δ

u (6-5o)

For , we find 16D

2 216 , , , , , , ,

1

2 2, , , , , , ,

1

1 [ ( )2

1 ( )2

k

y j y j x j x j x j y j y x jj

k

y j y j x j y j x j x j y jj

D A h u c h u

A h u A h c u

ψ=

=

= − Δ + Δ − Δ

= − Δ + Δ

2 ]

.

u (6-5p)

6.4 Summary

In this chapter, we have derived the primary wave aberration coefficients for cross

cylindrical anamorphic systems as:

Page 113: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

112

Primary wave aberration coefficients associated with x-RSOS

,21 , ,

1

18 8

kx j 1

x j x j Ixj j

uD A h

n=

= − Δ = −∑ S ,

,4 , , ,

1

12 2

kx j 1

x j x j x j IIxj j

uD A A h

n=

= − Δ = −∑ S ,

,2 28 , , ,

1

1 (3 ) (3 )4 4

kx j 1

x j x j x x j IIIx IVxj j

uD A h P S

n=

= − Δ + Ψ = − +∑ S ,

313 , , , , , , ,2

1

1 1[ ( ) ]2 2

k 1x j x j x j x j x x j x j x j Vx

j j

D A h A h A h Pn=

= − Δ + Ψ − = −∑ S .

Primary wave aberration coefficients associated with y-RSOS

,22 , ,

1

18 8

ky j

y j y j Iyj j

uD A h

n=

= − Δ = −∑ 1 S ,

,7 , , ,

1

12 2

ky j

y j y j y j IIyj j

uD A A h

n=

= − Δ = −∑ 1 S ,

,2 29 , , ,

1

1 (3 ) (3 )4 4

ky j

y j y j y y j IIIy IVyj j

uD A h P S

n=

= − Δ + Ψ = − +∑ 1 S ,

314 , , , , , , ,2

1

1 1[ ( )2 2

k

y j y j y j y j y y j y j y j Vyj j

D A h A h A h Pn=

= − Δ + Ψ − = −∑ 1] S .

Additional terms for skew rays

2 23 , , , , ,

1

1 (4

k

y j y j x j x j x j y jj

D A h u h c=

= − Δ + Δ∑ , )u ,

2 25 , , , , , ,

1

1 (2

k

y j y j x j y j x j x j y jj

D A h u A h c=

= − Δ + Δ∑ , )u ,

Page 114: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

113

6 , , , , , , ,1

1 (2

k

y j y j x j x j x j x j x j y jj

D A h u u h h c u=

= − Δ + Δ∑ , ) ,

2 210 , , , , , ,

1

1 (4

k

)x j x j y j y j y j x jj

D A h u h c=

= − Δ + Δ∑ u ,

2 211 , , , , , ,

1

1 (4

k

y j y j x j x j x j y jj

D A h u h c=

= − Δ + Δ∑ )u ,

12 , , , , , , , , ,1

(k

y j y j x j x j y j x j x j x j y jj

D A h u u A h h c u=

= − Δ + Δ∑ ) ,

2 215 , , , , , , ,

1

1 (2

k

)x j x j y j x j y j y j x jj

D A h u A h c=

= − Δ + Δ∑ u ,

2 216 , , , , , , ,

1

1 (2

k

y j y j x j y j x j x j y jj

D A h u A h c=

= − Δ + Δ∑ )u .

If we compare the results with what we got in chapter 5, we will see the primary

aberration coefficients for cross cylindrical anamorphic systems are more general than

those of parallel cylindrical anamorphic systems, and the former can be reduced into the

latter by taking all radii of curvature in one principal section as zero ( in chapter 5, it is

). This makes sense because parallel cylindrical anamorphic systems are special

cases of cross cylindrical anamorphic systems.

, 0x jc =

Thus, C. G. Wynne’s result [13] is just a special case of ours. Furthermore, our

treatment does not have any restriction on location of the system stop.

Page 115: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

114

CHAPTER 7

PRIMARY ABERRATION THEORY FOR TOROIDAL ANAMORPHIC

SYSTEMS

In chapters 5 and 6, the primary aberration coefficients for two types of cylindrical

anamorphic systems were obtained using the method we developed in chapter 2 through

chapter 4. Now, let us go a step further and consider another group of anamorphic

systems made from toroidal lenses.

To preserve structure and organization, and since we are applying the same method

onto different anamorphic system types, the development procedures in this chapter

resemble those in chapters 5 and 6.

This chapter presents the primary ray aberration coefficients for toroidal anamorphic

systems in section 7.1, and the primary wave aberration coefficients for toroidal

anamorphic systems and their simplifications in sections 7.2 and 7.3.

Notice that since a cylindrical surface is a special case of a toroidal surface with one

principal radius of curvature equal to infinity, all major results presented in this chapter

will be applicable to cylindrical anamorphic systems also.

7.1 The primary ray aberration coefficients for toroidal anamorphic systems

From chapter 2.3, we know the surface sag of the toroidal surface to a fourth-order

approximation is

Page 116: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

115

2 2 4 2 2 4

3 21 1 2( ) (2 8x y x x y y

3 )x y x x y yzr r r r r r

= + + + + . (7-1)

From equation (4-13d) and equations (4-14), we have

22 , 1 ,

1 (2

21)j x j j y j jz c x c y= + , (7-2a)

3 3 3 2 2, 1 , , 1 1

3

1 (2

j )x j j x j y j j jj

c x c c x yαγ

= − + , (7-2b)

3 3 3 2 2, 1 , , 1 1

3

1 (2

jy j j x j y j j j

j

c y c c x y )βγ

= − + . (7-2c)

By putting equations (7-2) and equations (4-19) into equations (4-17)), we get

3' ' ' 2, 3 , 1 2 2 1 2 , , 2 1

1 1 1 3

2 2 2 2, , , , , , ,

1

2 2 2 2, , , , ,

[ ( ) ( ) ]

1 [( )( 2 )2

1 [( )( 22

k k kj

k x k k x j j j j j x x j x j x j j j jj j j j

k

x j x x j x x j x x j x x j x x x j x jj

x j x x j x x j y y j y y j

n u A x N z L H N h c z x n

A h H h u H u H u u

A h H h u H u

αδξ δ ψ δ

γ

ρ ρ ρ

ρ ρ ρ

= = =

=

= Δ − + + − +

= − + Δ + Δ + Δ

− + Δ + Δ +

∑ ∑ ∑

, ,1

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , ,

1

)

1 [ ( 2 )( )]2

1 [ ( 2 )( )]2

1 ( 2 )2

k

y y y j y jj

k

x j x j x j x x j x j x x x j x x x j x x jj

k

x j y j x j y y j y j y y y j y x x j x x jj

x x x x j x x j x x x j x jj

H u u

A c h h h H h H u H u

A c h h h H h H u H u

H u H u H u u

ρ ρ ρ

ρ ρ ρ

ψ ρ ρ

=

=

=

=

Δ

− + + Δ + Δ

− + + Δ + Δ

+ Δ + Δ + Δ

2 2 2 2, , , ,

1

2 2, , , , , , , , , ,

1

3 3 2, , , , , , , , , ,

1

1 ( 2 )2

1 ( )[ ( ) (2

1 [ ( ) ( )(2

k

k

x x x y j y y j y y y j y jj

k2) ]x j x j x x j x x j x j x x j x x j y j y y j y y j j

j

k

x j x j x x j x x j x j y j x x j x x j y y j y yj

H u H u H u u

h c h H h c h H h c h H h n

h c h H h c c h H h h H h

ψ ρ ρ

ρ ρ ρ

ρ ρ ρ

=

=

=

+ Δ + Δ + Δ

− + + + +

+ + + + +

∑ 2) ]

Δ

j jnΔ

(7-3a)

Page 117: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

116

3' ' ' 2, 3 , 1 2 2 1 2 , , 2 1

1 1 1 3

2 2 2 2, , , , , , ,

1

2 2 2 2, , , , ,

[ ( ) ( ) ]

1 [( )( 2 )2

1 [( )( 22

k k kj

k y k k y j j j j j y y j y j y j j j jj j j j

k

y j y y j y y j x x j x x j x x x j x jj

y j y y j y y j y y j y y j

n u A y N z M H N h c z y n

A h H h u H u H u u

A h H h u H u

βδη δ ψ δ

γ

ρ ρ ρ

ρ ρ ρ

= = =

=

= Δ − + + − +

= − + Δ + Δ + Δ

− + Δ + Δ +

∑ ∑ ∑

, ,1

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , ,

1

)

1 [ ( 2 )( )]2

1 [ ( 2 )( )]2

1 ( 2 )2

k

y y y j y jj

k

y j x j x j x x j x j x x x j x y y j y y jj

k

y j y j y j y y j y j y y y j y y y j y y jj

y y x x j x x j x x x j x jj

H u u

A c h h h H h H u H u

A c h h h H h H u H u

H u H u H u u

ρ ρ ρ

ρ ρ ρ

ψ ρ ρ

=

=

=

=

Δ

− + + Δ + Δ

− + + Δ + Δ

+ Δ + Δ + Δ

2 2 2 2, , , ,

1

2 2, , , , , , , , , ,

1

3 3 2 2, , , , , , , , ,

1

1 ( 2 )2

1 ( )[ ( ) (2

1 [ ( ) ( ) (2

k

k

y y y y j y y j y y y j y jj

k

y j y j y y j y y j x j x x j x x j y j y y j y y j jj

k

y j y j y y j y y j x j y j x x j x x j y y j y yj

H u H u H u u

h c h H h c h H h c h H h n

h c h H h c c h H h h H h

ψ ρ ρ

ρ ρ ρ

ρ ρ ρ

=

=

=

+ Δ + Δ + Δ

− + + + +

+ + + + +

∑ , )]

2) ]Δ

j jnΔ

(7-3b)

By expanding and regrouping the above equations by their field and aperture

dependences, as we did in chapters 5 and 6, we get the corresponding primary ray

aberrations terms in a form similar to equations (3-8):

For Spherical Aberration-like aberration types

1 :D' ' ' 2 2 3

, 3 , , , , , ,1

' ' ', 3

1 [ ( )]2

0

k

k x k k x j x j x j x j x j x j xj

k y k k

n u A h u c h u

n u

δξ ρ

δη=

⎫= − Δ + Δ ⎪

⎬⎪= ⎭

∑ (7-4a)

2 :D

' ' ', 3

' ' ' 2 2 3, 3 , , , , , ,

1

0

1 [ ( )]2

k x k k

k

k y k k y j y j y j y j y j y j yj

n u

n u A h u c h u

δξ

δη ρ=

⎫=⎪⎬

= − Δ + Δ ⎪⎭

∑ (7-4b)

Page 118: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

117

3 :D

' ' ' 2 2 2, , , , , , ,

1

' ' ' 2 2, , , , , , ,

1

2 2 2 2 2, , , , , ,

1 [ ( )]2

1 {[ ( )]2

( ) }

k

k x k k x j x j y j y j y j x j x yj

k

k y k k y j y j x j x j x j y jj

x j y j x j y j x j y j j x y

n u A h u c h u

n u A h u c h u

c c c c h h n

δξ ρ ρ

δη

ρ ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ⎪⎪⎭

∑ (7-4c)

For Coma-like aberration types

4 :D

' ' ' 2 2, 3 , , , , , ,

1

2, , , , , , , ,

' ' ', 3

1 [ (2

2 2 ) ]

0

k

k x k k x j x j x j x j x j x jj

x j x j x j x j x j x j x j x x j x x

k y k k

n u A h u c h u

h u u c h h u u H

n u

δξ

ψ

δη

=

⎫= − Δ + Δ ⎪

⎪⎪+ Δ + Δ − Δ ⎬⎪

= ⎪⎪⎭

∑2ρ (7-4d)

5 :D

' ' ', 3 , , , , , , , ,

1

' ' ' 2 2, 3 , , , , , ,

1

2 2 2 2, , , , , , , ,

[ ( )]

1 [ ( )2

( ) ]

k

k x k k x j x j y j y j y j y j y j x j y x yj

k

k y k k y j y j x j x j x j y jj

y x j x j y j x j y j x j y j y j j y x

n u A h u u c h h u H

n u A h u c h u

u c c c c h h h n H 2

δξ ρ

δη

ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎪⎪= − Δ + Δ ⎬⎪⎪−Ψ Δ + − Δ⎪⎪⎭

ρ

(7-4e)

6 :D

' ' ' 2 2 2 2, 3 , , , , , , ,

1

' ' ', 3 , , , , , , , ,

1

2 2 2, , , , , , ,

1 [ ( ) ]2

[ ( )

( ) ]

k

k x k k x j x j y j y j y j x j x y j x yj

k

k y k k y j y j x j x j x j x j x j y jj

x j y j x j y j x j x j y j j x x y

n u A h u c h u u H

n u A h u u c h h u

c c c c h h h n H

δξ ρ

δη

ρ ρ

=

=

⎫= − Δ + Δ −Ψ Δ ⎪

⎪⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ⎪⎪⎭

∑ (7-4f)

7 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , , , , ,

1

2 2, , , , ,

0

1 [ ( 22

2 ) ]

k x k k

k

k y k k y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y y

n u

n u A h u c h u h u u

c h h u u H

δξ

δη

ρ=

⎫=⎪⎪= − Δ + Δ + Δ ⎬⎪⎪+ Δ −Ψ Δ ⎭

∑ (7-4g)

Page 119: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

118

For Astigmatism and Field curvature-like aberration types

8 :D

' ' ' 2 2, 3 , , , , , , , , ,

1

2, , , , , ,

' ' ', 3

1 [ ( 22

2 ) 2 ]

0

k

k x k k x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j x j x x

k y k k

n u A h u c h u h u u

c h h u u u H

n u

δξ

ψ ρ

δη

=

⎫= − Δ + Δ + Δ ⎪

⎪⎪+ Δ − Δ ⎬⎪

= ⎪⎪⎭

∑ (7-4h)

9 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , , , , ,

1

2, , , , , ,

0

1 [ ( 22

2 ) 2 ]

k x k k

k

k y k k y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j y y

n u

n u A h u c h u h u u

c h h u u u H

δξ

δη

ψ ρ=

⎫=⎪⎪= − Δ + Δ + Δ ⎬⎪⎪+ Δ − Δ ⎭

∑ (7-4i)

10 :D' ' ' 2 2 2

, 3 , , , , , ,1

' ' ', 3

1 [ ( )]2

0

k

k x k k x j x j y j y j y j x j y xj

k y k k

n u A h u c h u H

n u

δξ ρ

δη=

⎫= − Δ + Δ ⎪

⎬⎪= ⎭

∑ (7-4j)

11 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

2 2 2 2 2, , , , , ,

0

1 [ ( )2

( ) ]

k x k k

k

k y k k y j y j x j x j x j y jj

x j y j x j y j x j y j j x y

n u

n u A h u c h u

c c c c h h n H

δξ

δη

ρ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ ⎭

∑ (7-4k)

12 :D

' ' ', 3 , , , , , , , ,

1

, ,

' ' ', 3 , , , , , , , , , ,

1

2 2, , , , , , , ,

[ ( )

]

[ ( )

( ) ]

k

k x k k x j x j y j y j y j y j y j x jj

x y j y j x y y

k

k y k k y j y j x j x j x j x j x j y j y x j x jj

x j y j x j y j x j x j y j y j j x y x

n u A h u u c h h u

u u H H

n u A h u u c h h u u u

c c c c h h h h n H H

δξ

ψ ρ

δη ψ

ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎪− Δ⎬

= − Δ + Δ − Δ

+ − Δ

∑⎪

⎪⎪⎪⎪⎭

(7-4 ) l

For Distortion-like aberration types

13 :D' ' ' 2 2 2 3

, 3 , , , , , , ,1

' ' ', 3

1 [ ( ) ]2

0

k

k x k k x j x j x j x j x j x j x x j xj

k y k k

n u A h u c h u u H

n u

δξ ψ

δη=

⎫= − Δ + Δ − Δ ⎪

⎬⎪= ⎭

∑ (7-4m)

Page 120: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

119

14 :D

' ' ', 3

' ' ' 2 2 2 3, 3 , , , , , , ,

1

0

1 [ ( ) ]2

k x k k

k

k y k k y j y j y j y j y j y j y y j yj

n u

n u A h u c h u u H

δξ

δη ψ=

⎫=⎪⎬

= − Δ + Δ − Δ ⎪⎭

∑ (7-4n)

15 :D' ' ' 2 2 2 2

, 3 , , , , , , ,1

' ' ', 3

1 [ ( ) ]2

0

k

k x k k x j x j y j y j y j x j x y j x yj

k y k k

n u A h u c h u u H H

n u

δξ ψ

δη=

⎫= − Δ + Δ − Δ ⎪

⎬⎪= ⎭

∑ (7-4o)

16 :D

' ' ', 3

' ' ' 2 2 2, 3 , , , , , , ,

1

2 2 2 2, , , , , , ,

0

1 [ ( )2

( ) ]

k x k k

k

k y k k y j y j x j x j x j y j y x jj

x j y j x j y j x j y j y j j x y

n u

n u A h u c h u u

c c c c h h h n H H

δξ

δη ψ=

⎫=⎪⎪= − Δ + Δ − Δ ⎬⎪⎪+ − Δ ⎭

∑ (7-4p)

From chapter 6.1, we know in cross cylindrical anamorphic systems, for any

surface j , either ,x jc or ,y jc will be zero, thus any term containing , ,x j yc c× j will be zero.

With this point in mind, by taking 2, , 0x j y jc c = and 2

, , 0x j y jc c = , we can reduce equations

(7-4) into equations (6-3).

This means cylindrical anamorphic systems are subgroups of toroidal anamorphic

systems and all equations developed in this chapter will be valid for cylindrical

anamorphic system also.

7.2 The primary wave aberration coefficients for toroidal anamorphic systems

By comparing equations (7-4) with equations (3-8), we immediately get the primary

wave aberration coefficients though as 1D 16D

2 21 , , , , ,

1

1 (8

k

, )x j x j x j x j x j x jj

D A h u c h=

= − Δ + Δ∑ u (7-5a)

Page 121: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

120

2 22 , , , , ,

1

1 (8

k

y j y j y j y j y j y jj

D A h u c h=

= − Δ + Δ∑ , )u (7-5b)

2 23 , , , , ,

1

1 [ ( )4

k

, ]x j x j y j y j y j x jj

D A h u c h u=

= − Δ + Δ∑ (7-5c)

2 2

4 , , , , , , ,1

2, , , , ,

1 [ ( 26

2 ) ]

k

, ,x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j

D A h u c h u h u

c h h u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ u (7-5d)

5 , , , , , , , ,1

1 [ ( )2

k

]x j x j y j y j y j y j y j x jj

D A h u u c h h u=

= − Δ + Δ∑ (7-5e)

2 26 , , , , , ,

1

1 [ ( )2

k2

, ]x j x j y j y j y j x j x y jj

D A h u c h u=

= − Δ + Δ −Ψ Δ∑ u (7-5f)

2 2

7 , , , , , , ,1

2, , , , ,

1 [ ( 26

2 ) ]

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j

D A h u c h u h u

c h h u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ , ,u (7-5g)

2 2

8 , , , , , , ,1

, , , , , ,

1 [ ( 24

2 ) 2 ]

k

, ,x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j x j

D A h u c h u h u

c h h u u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ u (7-5h)

2 2

9 , , , , , , ,1

, , , , , ,

1 [ ( 24

2 ) 2 ]

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j

D A h u c h u h u

c h h u u uψ=

= − Δ + Δ + Δ

+ Δ − Δ

∑ , ,u (7-5i)

2 210 , , , , , ,

1

1 (4

k

)x j x j y j y j y j x jj

D A h u c h=

= − Δ + Δ∑ u (7-5j)

2 2

11 , , , , , ,1

2 2 2 2, , , , , ,

1 [ (4

( ) ]

k

y j y j x j x j x j y jj

x j y j x j y j x j y j j

D A h u c h

c c c c h h n=

= − Δ + Δ

+ − Δ

∑ )u (7-5k)

Page 122: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

121

12 , , , , , , , , , ,1

[ ( )k

]x j x j y j y j y j y j y j x j x y j y jj

D A h u u c h h u uψ=

= − Δ + Δ − Δ∑ u (7-5l)

2 213 , , , , , , ,

1

1 [ ( )2

k2 ]x j x j x j x j x j x j x x j

jD A h u c h u ψ

=

= − Δ + Δ − Δ∑ u (7-5m)

2 214 , , , , , , ,

1

1 [ ( )2

k

y j y j y j y j y j y j y y jj

D A h u c h u ψ=

= − Δ + Δ − Δ∑ 2 ]u (7-5n)

2 215 , , , , , , ,

1

1 [ ( )2

k2 ]x j x j y j y j y j x j x y j

jD A h u c h u ψ

=

= − Δ + Δ − Δ∑ u (7-5o)

2 2

16 , , , , , , ,1

2 2 2, , , , , , ,

1 [ ( )2

( ) ]

k

y j y j x j x j x j y j y x jj

x j y j x j y j x j y j y j j

D A h u c h u

c c c c h h h n

ψ=

= − Δ + Δ − Δ

+ − Δ

∑ 2u (7-5p)

Again, for , we can either choose the coefficient in the x-ray error

expression or the coefficient in the y-ray error expression. Their numerical values may

differ for any single surface

3 5 6 12, , ,D D D D

j , but the summation in the final image space is the same. In

our treatment above, the coefficient in the x-ray error expression was chosen because it

contains fewer terms.

7.3 Simplification of the results

We now have all of the primary wave aberration coefficients for a toroidal

anamorphic system in terms of the paraxial marginal and chief rays tracing data in both

symmetry planes. Again, let us use the corresponding paraxial definitions from chapter

2.11 to simplify the results we have obtained so that we can rewrite equations (7-5) in a

form as similar to the Seidel aberrations for RSOS as possible.

Page 123: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

122

For , we find 1D

2 21 , , , , ,

1

, , , , , ,1

,2, ,

1

1 ( )8

1 ( )8

181 .8

k

,x j x j x j x j x j x jj

k

x j x j x j x j x j x jj

kx j

x j x jj j

Ix

D A h u c h

A h u u h c

uA h

n

S

=

=

=

= − Δ + Δ

= − Δ +

= − Δ

= −

u

(7-6a)

For , we find 2D

2 22 , , , , ,

1

, , , , , ,1

,2, ,

1

1 ( )8

1 ( )8

181 .8

k

y j y j y j y j y j y jj

k

y j y j y j y j y j y jj

ky j

y j y jj j

Iy

D A h u c h

A h u u c h

uA h

n

S

=

=

=

= − Δ + Δ

= − Δ +

= − Δ

= −

,u

(7-6b)

For , we will keep it unchanged as 3D

2 23 , , , , ,

1

1 (4

k

, ).x j x j y j y j y j x jj

D A h u h c u=

= − Δ + Δ∑ (7-6c)

For , by comparison with equation (6-4d), we know 4D

2 24 , , , , , , ,

1

2, , , , ,

,, , ,

1

1 [ ( 26

2 ) ]

121 .2

k

, ,x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j

kx j

x j x j x jj j

IIx

D A h u c h u h u

c h h u uu

A A hn

S

ψ=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ

= −

u

(7-6d)

Page 124: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

123

For , we will keep it unchanged as 5D

5 , , , , , , ,1

1 (2

k

, ).x j x j y j y j y j y j y j x jj

D A h u u h h c u=

= − Δ + Δ∑ (7-6e)

For , we find 6D

2 26 , , , , , ,

1

2 2, , , , , , ,

1

1 [ ( )21 ( )2

k2

, ]

.

x j x j y j y j y j x j x y jj

k

x j x j y j x j y j y j x jj

D A h u c h u

A h u A h c u

=

=

= − Δ + Δ −Ψ Δ

= − Δ + Δ

u (7-6f)

For , by comparison with equations (6-5g) through (6-5i), we obtain 7 8 9, ,D D D

2 27 , , , , , , ,

1

2, , , , ,

,, , ,

1

1 [ ( 26

2 ) ]

121 .2

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j

ky j

y j y j y jj j

IIy

D A h u c h u h u

c h h u uu

A A hn

S

ψ=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ

= −

, ,u

(7-6g)

2 28 , , , , , , ,

1

, , , , , ,

,2 2, , ,

1

1 [ ( 24

2 ) 2 ]

1 (3 )41 (3 ).4

k

, ,x j x j x j x j x j x j x j x j xj

x j x j x j x j x x j x j

kx j

x j x j x x jj j

IIIx IVx

D A h u c h u h u

c h h u u uu

A h Pn

S S

ψ=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ + Ψ

= − +

ju

(7-6h)

2 29 , , , , , , ,

1

, , , , , ,

,2 2, , ,

1

1 [ ( 24

2 ) 2 ]

1 (3 )41 (3 ).4

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j

ky j

y j y j y y jj j

IIIy IVy

D A h u c h u h u

c h h u u uu

A h Pn

S S

ψ=

=

= − Δ + Δ + Δ

+ Δ − Δ

= − Δ + Ψ

= − +

, ,u

(7-6i)

Page 125: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

124

For and , no change is needed, as 10D 11D

2 210 , , , , , ,

1

1 (4

k

).x j x j y j y j y j x jj

D A h u h c=

= − Δ + Δ∑ u (7-6k)

2 2 2 2 2 211 , , , , , , , , , , , ,

1 1

1 1( ) ( )4 4

k k

y j y j x j x j x j y j x j y j x j y j x j y j jj j

D A h u h c u c c c c h h n= =

= − Δ + Δ − − Δ∑ ∑ . (7-6k)

For , we find 12D

12 , , , , , , , , , ,

1

, , , , , , , , ,1

[ ( )

( )

k

]

.

x j x j y j y j y j y j y j x j x y j y jj

k

x j x j y j y j x j y j y j y j x jj

D A h u u c h h u u

A h u u A h h c u

ψ=

=

= − Δ + Δ − Δ

= − Δ + Δ

u (7-6 l )

For , , by comparing with equation (6-5m) and (6-5n), we immediately get 13D 14D

2 213 , , , , , , ,

1

3, , , , , , ,2

1

1 [ ( )2

1 1[ (21 .2

k2 ]

) ]

x j x j x j x j x j x j x x jj

k

x j x j x j x j x x j x j x jj j

Vx

D A h u c h u

A h A h A h Pn

S

ψ=

=

= − Δ + Δ − Δ

= − Δ + Ψ −

= −

u

(7-6m)

2 2 214 , , , , , , ,

1

3, , , , , , ,2

1

1 [ ( ) ]2

1 1[ (21 .2

k

y j y j y j y j y j y j y y jj

k

y j y j y j y j y y j y j y jj j

Vy

D A h u c h u

A h A h A h Pn

S

ψ=

=

= − Δ + Δ − Δ

= − Δ + Ψ −

= −

∑ ) ]

u

(7-6n)

For , we find 15D

2 215 , , , , , , ,

1

2 2, , , , , , ,

1

1 [ ( )21 ( )2

k2 ]x j x j y j y j y j x j x y j

j

k

x j x j y j x j y j y j x jj

D A h u c h u

A h u A h c u

ψ=

=

= − Δ + Δ − Δ

= − Δ + Δ

u (7-6o)

Page 126: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

125

For , we find 16D

2 216 , , , , , , ,

1

2 2 2, , , , , , ,

2 2, , , , , , ,

1

2 2 2, , , , , , ,

1

1 [ ( )2

( ) ]

1 ( )2

1 ( ) .2

k

y j y j x j x j x j y j y x jj

x j y j x j y j x j y j y j j

k

y j y j x j y j x j x j y jj

k

x j y j x j y j x j y j y j jj

D A h u c h u

c c c c h h h n

A h u A h c u

c c c c h h h n

ψ=

=

=

= − Δ + Δ − Δ

+ − Δ

= − Δ + Δ

− − Δ

2u

(7-6p)

7.4 Summary

In this chapter, we have found the primary wave aberration coefficients for toroidal

anamorphic systems in subgroups as:

Primary wave aberration coefficients associated with x-RSOS

,21 , ,

1

18 8

kx j 1

x j x j Ixj j

uD A h

n=

= − Δ = −∑ S

,4 , , ,

1

12 2

kx j 1

x j x j x j IIxj j

uD A A h

n=

= − Δ = −∑ S

,2 28 , , ,

1

1 (3 ) (3 )4 4

kx j 1

x j x j x x j IIIx IVxj j

uD A h P S

n=

= − Δ + Ψ = − +∑ S

313 , , , , , , ,2

1

1 1[ ( ) ]2 2

k 1x j x j x j x j x x j x j x j Vx

j j

D A h A h A h Pn=

= − Δ + Ψ − = −∑ S

Primary wave aberration coefficients associated with y-RSOS

,22 , ,

1

18 8

ky j

y j y j Iyj j

uD A h

n=

= − Δ = −∑ 1 S

,7 , , ,

1

12 2

ky j

y j y j y j IIyj j

uD A A h

n=

= − Δ = −∑ 1 S

Page 127: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

126

,2 29 , , ,

1

1 (3 ) (3 )4 4

ky j

y j y j y y j IIIy IVyj j

uD A h P S

n=

= − Δ + Ψ = − +∑ 1 S

314 , , , , , , ,2

1

1 1[ ( )2 2

k

y j y j y j y j y y j y j y j Vyj j

D A h A h A h Pn=

= − Δ + Ψ − = −∑ 1] S

Additional terms for skew rays

2 23 , , , , ,

1

1 (4

k

, )x j x j y j y j y j x jj

D A h u h c=

= − Δ + Δ∑ u

5 , , , , , , ,1

1 (2

k

, )x j x j y j y j y j y j y j x jj

D A h u u h h c u=

= − Δ + Δ∑

2 26 , , , , , ,

1

1 (2

k

, )x j x j y j x j y j y j x jj

D A h u A h c=

= − Δ + Δ∑ u

2 210 , , , , , ,

1

1 (4

k

)x j x j y j y j y j x jj

D A h u h c=

= − Δ + Δ∑ u

2 2 2 2 2 211 , , , , , , , , , , , ,

1 1

1 1( ) ( )4 4

k k

y j y j x j x j x j y j x j y j x j y j x j y j jj j

D A h u h c u c c c c h h n= =

= − Δ + Δ − − Δ∑ ∑

12 , , , , , , , , ,1

(k

)x j x j y j y j x j y j y j y j x jj

D A h u u A h h c u=

= − Δ + Δ∑

2 215 , , , , , , ,

1

1 (2

k

)x j x j y j x j y j y j x jj

D A h u A h c=

= − Δ + Δ∑ u

2 2 2 2 216 , , , , , , , , , , , , , ,

1 1

1 1( ) ( )2 2

k k

y j y j x j y j x j x j y j x j y j x j y j x j y j y j jj j

D A h u A h c u c c c c h h h n= =

= − Δ + Δ − − Δ∑ ∑

If we compare the results with what we got in chapter 6, we will see the primary

aberration coefficients for the two associated RSOS are the same for both anamorphic

system types. However, the primary aberration coefficients for the skew rays are different.

Page 128: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

127

CHAPTER 8

PRIMARY ABERRATION THEORY FOR GENERAL ANAMORPHIC

SYSTEMS

Chapters 5, 6, and 7 give the primary aberration coefficients for anamorphic systems

made from specified types of surfaces, such as cylindrical surfaces and toroidal surfaces.

In this chapter, we will consider the general anamorphic systems constructed with

general double curvature surface type, with the allowance of fourth-order aspheric

departures in the two principal sections.

Following the same development pattern as used from chapters 5 through 7, we will

present the primary ray aberration coefficients for general anamorphic systems in section

8.1, and the primary wave aberration coefficients for general anamorphic systems and

their simplifications in sections 8.2 and 8.3.

8.1 The primary ray aberration coefficients for general anamorphic systems

From chapter 2.3, we know in general that we can write the sag equation for any

double curvature surface with a fourth-order approximation as

2 2 4 2 2 4

3 33 4 5

1 1 2( ) (2 8x y

3 )x y x x yzr r r r r

= + + + +y . (8-1)

Notice that in this equation, we permit the fourth-order aspheric departure in both

principal sections by allowing to be different from3 4 5, ,r r r ,x yr r or their combinations.

In this case, from equation (4-13d) and equations (4-14), we have

Page 129: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

128

22 , 1 ,

1 (2j x j j y j j

21 )z c x c y= + , (8-2a)

3 2

3 1 13

3 3, 4,

(2 2

j j j

j j

13 )j

j

x x yr r

αγ

= − + , (8-2b)

3 2

3 1 13

3 5, 4,

(2 2

j j j

j j

y x yr r

βγ

= − + 13 )j

j

. (8-2c)

By putting equations (8-2) together with equations (4-19) into the anamorphic

primary ray aberration equations (4-17a) and (4-17b), we get

3' ' ' 2, 3 , 1 2 2 1 2 , , 2 1

1 1 1 3

2 2 2 2, , , , , , ,

1

2 2 2 2, , , , ,

[ ( ) ( ) ]

1 [( )( 2 )2

1 [( )( 22

k k kj

k x k k x j j j j j x x j x j x j j j jj j j j

k

x j x x j x x j x x j x x j x x x j x jj

x j x x j x x j y y j y y j

n u A x N z L H N h c z x n

A h H h u H u H u u

A h H h u H u

αδξ δ ψ δ

γ

ρ ρ ρ

ρ ρ ρ

= = =

=

= Δ − + + − +

= − + Δ + Δ + Δ

− + Δ + Δ +

∑ ∑ ∑

, ,1

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , ,

1

)

1 [ ( 2 )( )]2

1 [ ( 2 )( )]2

1 ( 2 )2

k

y y y j y jj

k

x j x j x j x x j x j x x x j x x x j x x jj

k

x j y j y j y y j y j y y y j y x x j x x jj

x x x x j x x j x x x j x jj

H u u

A c h h h H h H u H u

A c h h h H h H u H u

H u H u H u u

ρ ρ ρ

ρ ρ ρ

ψ ρ ρ

=

=

=

=

Δ

− + + Δ + Δ

− + + Δ + Δ

+ Δ + Δ + Δ

2 2 2 2, , , ,

1

2 2, , , , , , , , , ,

1

3 3 3 2, 3, , , 4, , , , ,

1

1 ( 2 )2

1 ( )[ ( ) (2

1 [ ( ) ( )( ) ]2

k

k

x x y y j y y j y y y j y jj

k2) ]x j x j x x j x x j x j x x j x x j y j y y j y y j j

j

k

x j j x x j x x j j x x j x x j y y j y y jj

H u H u H u u

h c h H h c h H h c h H h n

h c h H h c h H h h H h

ψ ρ ρ

ρ ρ ρ

ρ ρ ρ

=

=

=

+ Δ + Δ + Δ

− + + + +

+ + + + +

∑ jnΔ

Δ

(8-3a)

Page 130: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

129

3' ' ' 2, 3 , 1 2 2 1 2 , , 2 1

1 1 1 3

2 2 2 2, , , , , , ,

1

2 2 2 2, , , , ,

[ ( ) ( ) ]

1 [( )( 2 )2

1 [( )( 22

k k kj

k y k k y j j j j j y y j y j y j j j jj j j j

k

y j y y j y y j x x j x x j x x x j x jj

y j y y j y y j y y j y y j

n u A y N z M H N h c z y n

A h H h u H u H u u

A h H h u H u

βδη δ ψ δ

γ

ρ ρ ρ

ρ ρ ρ

= = =

=

= Δ − + + − +

= − + Δ + Δ + Δ

− + Δ + Δ +

∑ ∑ ∑

, ,1

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , , , , , ,

1

2 2 2 2, , , ,

1

)

1 [ ( 2 )( )]2

1 [ ( 2 )( )]2

1 ( 2 )2

k

y y y j y jj

k

y j x j x j x x j x j x x x j x y y j y y jj

k

y j y j y j y y j y j y y y j y y y j y y jj

y y x x j x x j x x x j x jj

H u u

A c h h h H h H u H u

A c h h h H h H u H u

H u H u H u u

ρ ρ ρ

ρ ρ ρ

ψ ρ ρ

=

=

=

=

Δ

− + + Δ + Δ

− + + Δ + Δ

+ Δ + Δ + Δ

2 2 2 2, , , ,

1

2 2, , , , , , , , , ,

1

3 3 3 2, 5, , , 4, , , , ,

1

1 ( 2 )2

1 ( )[ ( ) (2

1 [ ( ) ( ) ( )]2

k

k

y y y y j y y j y y y j y jj

k

y j y j y y j y y j x j x x j x x j y j y y j y y j jj

k

y j j y y j y y j j x x j x x j y y j y y jj

H u H u H u u

h c h H h c h H h c h H h n

h c h H h c h H h h H h

ψ ρ ρ

ρ ρ ρ

ρ ρ ρ

=

=

=

+ Δ + Δ + Δ

− + + + +

+ + + + +

∑ jnΔ

2) ]Δ

(8-3b)

By expanding and regrouping the results according to their field and aperture

dependences, as we did in chapters 5, 6 and 7, we obtain the corresponding primary ray

aberration terms in a form similar to equations (3-8) as:

For Spherical Aberration-like aberration types

1 :D

' ' ' 2 2, 3 , , , , , ,

1

3 3 4 3, 3, ,

' ' ', 3

1 {[ ( )]2

( ) }

0

k

k x k k x j x j x j x j x j x jj

x j j x j j x

k y k k

n u A h u c h u

c c h n

n u

δξ

ρ

δη

=

⎫= − Δ + Δ ⎪

⎪⎪+ − Δ ⎬⎪

= ⎪⎪⎭

∑ (8-4a)

Page 131: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

130

2 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

3 3 4 3, 5, ,

0

1 {[ ( )]2

( ) }

k x k k

k

k y k k y j y j y j y j y j y jj

y j j y j j y

n u

n u A h u c h u

c c h n

δξ

δη

ρ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ ⎭

∑ (8-4b)

3 :D

' ' ' 2 2, , , , , ,

1

2 3 2 2 2, , 4, , ,

' ' ' 2 2, , , , , ,

1

2 3 2 2 2, , 4, , ,

1 {[ ( )]2

( ) }

1 {[ ( )]2

( ) }

k

k x k k x j x j y j y j y j x jj

x j y j j x j y j j x y

k

k y k k y j y j x j x j x j y jj

x j y j j x j y j j x y

n u A h u c h u

c c c h h n

n u A h u c h u

c c c h h n

δξ

ρ ρ

δη

ρ ρ

=

=

⎫= − Δ + Δ ⎪

⎪⎪+ − Δ ⎪⎬⎪= − Δ + Δ⎪⎪

+ − Δ ⎪⎭

,

,

(8-4c)

For Coma-like aberration types

4 :D

' ' ' 2 2, 3 , , , , , , , , ,

1

2 3 3 3, , , , , , 3, , ,

' ' ', 3

1 [ ( 22

2 ) 3( ) ]

0

k

k x k k x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j x j j x j x j j x x

k y k k

n u A h u c h u h u u

c h h u u c c h h n H

n u

δξ

ψ ρ

δη

=

⎫= − Δ + Δ + Δ

2

⎪⎪⎪+ Δ − Δ + − Δ ⎬⎪

= ⎪⎪⎭

∑(8-4d)

5 :D

' ' ', 3 , , , , , , , ,

1

2 3 2, , 4, , , ,

' ' ' 2 2, 3 , , , , , ,

1

2 2 3 2, , , 4, , , ,

[ ( )

( ) ]

1 [ ( )2

( ) ]

k

k x k k x j x j y j y j y j y j y j x jj

x j y j j x j y j y j j y x y

k

k y k k y j y j x j x j x j y jj

y x j x j y j j x j y j y j j y

n u A h u u c h h u

c c c h h h n H

n u A h u c h u

u c c c h h h n H

δξ

ρ ρ

δη

ρ

=

=

= − Δ + Δ

+ − Δ

= − Δ + Δ

−Ψ Δ + − Δ

∑2

x

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(8-4e)

6 :D

' ' ' 2 2, 3 , , , , , ,

1

2 2 3 2, , , 4, , , ,

' ' ', 3 , , , , , , , ,

1

2 3 2, , 4, , , ,

1 [ ( )2

( ) ]

[ ( )

( ) ]

k

k x k k x j x j y j y j y j x jj

2x y j x j y j j x j x j y j j x y

k

k y k k y j y j x j x j x j x j x j y jj

x j y j j x j x j y j j x x

n u A h u c h u

u c c c h h h n H

n u A h u u c h h u

c c c h h h n H

δξ

ρ

δη

ρ

=

=

= − Δ + Δ

−Ψ Δ + − Δ

= − Δ + Δ

+ − Δ

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(8-4f)

Page 132: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

131

7 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , , , , ,

1

2 3 3 3, , , , , , 5, , ,

0

1 [ ( 22

2 ) 3( ) ]

k x k k

k

k y k k y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j j y j y j j y y

n u

n u A h u c h u h u u

c h h u u c c h h n H

δξ

δη

ρ=

⎫=

2

⎪⎪= − Δ + Δ + Δ ⎬⎪⎪+ Δ −Ψ Δ + − Δ ⎭

∑ (8-4g)

For Astigmatism and Field curvature-like aberration types

8 :D

' ' ' 2 2, 3 , , , , , , , , ,

1

3 3 2 2 2, , , , , , , 3, , ,

' ' ', 3

1 [ ( 22

2 ) 2 3( ) ]

0

k

k x k k x j x j x j x j x j x j x j x j x jj

x j x j x j x j x x j x j x j j x j x j j x x

k y k k

n u A h u c h u h u u

c h h u u u c c h h n H

n u

δξ

ψ ρ

δη

=

⎫= − Δ + Δ + Δ ⎪

⎪⎪+ Δ − Δ + − Δ ⎬⎪

= ⎪⎪⎭

∑(8-4h)

9 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , , , , ,

1

3 3 2 2 2, , , , , , , 5, , ,

0

1 [ ( 22

2 ) 2 3( ) ]

k x k k

k

k y k k y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j y j j y j y j j y y

n u

n u A h u c h u h u u

c h h u u u c c h h n H

δξ

δη

ψ ρ=

⎫=⎪⎪= − Δ + Δ + Δ ⎬⎪⎪+ Δ − Δ + − Δ ⎭

∑ (8-4i)

10 :D

' ' ' 2 2, 3 , , , , , ,

1

2 3 2 2 2, , 4, , ,

' ' ', 3

1 [ ( )2

( ) ]

0

k

k x k k x j x j y j y j y j x jj

x j y j j x j y j j y x

k y k k

n u A h u c h u

c c c h h n H

n u

δξ

ρ

δη

=

⎫= − Δ + Δ ⎪

⎪⎪+ − Δ ⎬⎪

= ⎪⎪⎭

∑ (8-4j)

11 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

2 3 2 2 2, , 4, , ,

0

1 [ ( )2

( ) ]

k x k k

k

k y k k y j y j x j x j x j y jj

x j y j j x j y j j x y

n u

n u A h u c h u

c c c h h n H

δξ

δη

ρ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪+ − Δ ⎭

∑ (8-4k)

Page 133: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

132

12 :D

' ' ', 3 , , , , , , , ,

1

2 3, , , , 4, , , , ,

' ' ', 3 , , , , , , , ,

1

2, , , , 4,

[ ( )

( ) ]

[ ( )

(

k

k x k k x j x j y j y j y j y j y j x jj

x y j y j x j y j j x j x j y j y j j x y y

k

k y k k y j y j x j x j x j x j x j y jj

y x j x j x j y j j

n u A h u u c h h u

u u c c c h h h h n H H

n u A h u u c h h u

u u c c c

δξ

ψ ρ

δη

ψ

=

=

= − Δ + Δ

− Δ + − Δ

= − Δ + Δ

− Δ + −

∑3

, , , ,) ]x j x j y j y j j x y xh h h h n H H ρ

⎫⎪⎪⎪⎪⎬⎪⎪⎪

Δ ⎪⎭

(8-4 ) l

For Distortion-like aberration types

13 :D

' ' ' 2 2, 3 , , , , , ,

1

2 3 3 3 3, , 3, , ,

' ' ', 3

1 [ ( )2

( ) ]

0

k

k x k k x j x j x j x j x j x jj

x x j x j j x j x j j x

k y k k

n u A h u c h u

u c c h h n H

n u

δξ

ψ

δη

=

⎫= − Δ + Δ ⎪

⎪⎪− Δ + − Δ ⎬⎪

= ⎪⎪⎭

∑ (8-4m)

14 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

2 3 3 3 3, , 5, , ,

0

1 [ ( )2

( ) ]

k x k k

k

k y k k y j y j y j y j y j y jj

y y j y j j y j y j j y

n u

n u A h u c h u

u c c h h n H

δξ

δη

ψ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪− Δ + − Δ ⎭

∑ (8-4n)

15 :D

' ' ' 2 2, 3 , , , , , ,

1

2 2 3 2 2, , , 4, , , ,

' ' ', 3

1 [ ( )2

( ) ]

0

k

k x k k x j x j y j y j y j x jj

x y j x j y j j x j x j y j j x y

k y k k

n u A h u c h u

u c c c h h h n H H

n u

δξ

ψ

δη

=

⎫= − Δ + Δ ⎪

⎪⎪− Δ + − Δ ⎬⎪

= ⎪⎪⎭

∑ (8-4o)

16 :D

' ' ', 3

' ' ' 2 2, 3 , , , , , ,

1

2 2 3 2 2, , , 4, , , ,

0

1 [ ( )2

( ) ]

k x k k

k

k y k k y j y j x j x j x j y jj

y x j x j y j j x j y j y j j x y

n u

n u A h u c h u

u c c c h h h n H H

δξ

δη

ψ=

⎫=⎪⎪= − Δ + Δ ⎬⎪⎪− Δ + − Δ ⎭

∑ (8-4p)

Page 134: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

133

By taking , we can reduce equations (8-4) into

equations (7-4). This makes sense because toroidal anamorphic systems are special cases

of general anamorphic systems.

3 23, , 4, , , 5, ,, ,j x j j x j y j j yc c c c c c c= = = j

8.2 The primary wave aberration coefficients for general anamorphic systems

By comparing equations (8-4) with equations (3-8), we immediately obtain the

primary wave aberration coefficients though as 1D 16D

2 2 3 3 41 , , , , , , , 3, ,

1

1 {[ ( )] ( ) }8

k

x j x j x j x j x j x j x j j x j jj

D A h u c h u c c h=

= − Δ + Δ + − Δ∑ n (8-5a)

2 2 3 3 42 , , , , , , , 5, ,

1

1 {[ ( )] ( ) }8

k

y j y j y j y j y j y j y j j y j jj

D A h u c h u c c h=

= − Δ + Δ + − Δ∑ n (8-5b)

2 2 2 3 2 23 , , , , , , , , 4, , ,

1

1 {[ ( )] ( ) }4

k

x j x j y j y j y j x j x j y j j x j y j jj

D A h u c h u c c c h h=

= − Δ + Δ + − Δ∑ n (8-5c)

2 2

4 , , , , , , , ,1

2 3 3 3, , , , , , 3, , ,

1 [ ( 26

2 ) 3( )

k

x j x j x j x j x j x j x j x j x jj

,

]x j x j x j x j x x j x j j x j x j j

D A h u c h u h u u

c h h u u c c h h nψ=

= − Δ + Δ + Δ

+ Δ − Δ + −

∑Δ

(8-5d)

5 , , , , , , , ,

1

2 3 2, , 4, , , ,

1 [ (2

( ) ]

k

)x j x j y j y j y j y j y j x jj

x j y j j x j y j y j j

D A h u u c h h

c c c h h h n=

= − Δ + Δ

+ − Δ

∑ u (8-5e)

2 2

6 , , , , , ,1

2 3 2, , 4, , , ,

1 [ ( )2

( ) ]

k2

,x j x j y j y j y j x j x y jj

x j y j j x j x j y j j

D A h u c h u

c c c h h h n=

= − Δ + Δ −Ψ Δ

+ − Δ

∑ u (8-5f)

Page 135: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

134

2 2

7 , , , , , , , ,1

2 3 3 3, , , , , , 5, , ,

1 [ ( 26

2 ) 3( )

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j j y j y j j

D A h u c h u h u u

c h h u u c c h h n=

= − Δ + Δ + Δ

+ Δ −Ψ Δ + −

∑ ,

(8-5g)

2 2

8 , , , , , , , , ,1

3 3 2 2, , , , , , , 3, , ,

1 [ ( 24

2 ) 2 3( )

k

x j x j x j x j x j x j x j x j x jj

]x j x j x j x j x x j x j x j j x j x j j

D A h u c h u h u u

c h h u u u c c h h nψ=

= − Δ + Δ + Δ

+ Δ − Δ + −

∑Δ

(8-5h)

2 2

9 , , , , , , , , ,1

3 3 2 2, , , , , , , 5, , ,

1 [ ( 24

2 ) 2 3( )

k

y j y j y j y j y j y j y j y j y jj

y j y j y j y j y y j y j y j j y j y j j

D A h u c h u h u u

c h h u u u c c h h nψ=

= − Δ + Δ + Δ

+ Δ − Δ + −

∑]Δ

(8-5i)

2 2 2 3 2 210 , , , , , , , , 4, , ,

1

1 [ ( ) ( )4

k

]x j x j y j y j y j x j x j y j j x j y j jj

D A h u c h u c c c h h=

= − Δ + Δ + − Δ∑ n (8-5j)

2 2 2 3 2 211 , , , , , , , , 4, , ,

1

1 [ ( ) ( )4

k

y j y j x j x j x j y j x j y j j x j y j jj

D A h u c h u c c c h h=

= − Δ + Δ + − Δ∑ ]n (8-5k)

12 , , , , , , , , , ,

1

2 3, , 4, , , , ,

[ ( )

( ) ]

k

x j x j y j y j y j y j y j x j x y j y jj

x j y j j x j x j y j y j j

D A h u u c h h u u

c c c h h h h n

ψ=

= − Δ + Δ − Δ

+ − Δ

∑ u (8-5 l )

2 2

13 , , , , , , ,1

3 3 3, 3, , ,

1 [ ( )2

( ) ]

k2

x j x j x j x j x j x j x x jj

x j j x j x j j

D A h u c h u

c c h h n

ψ=

= − Δ + Δ − Δ

+ − Δ

∑ u (8-5m)

2 2

14 , , , , , , ,1

3 3 3, 5, , ,

1 [ ( )2

( ) ]

k

y j y j y j y j y j y j y y jj

y j j y j y j j

D A h u c h u

c c h h n

ψ=

= − Δ + Δ − Δ

+ − Δ

∑ 2u (8-5n)

2 2

15 , , , , , , ,1

2 3 2, , 4, , , ,

1 [ ( )2

( ) ]

k2

x j x j y j y j y j x j x y jj

x j y j j x j x j y j j

D A h u c h u

c c c h h h n

ψ=

= − Δ + Δ − Δ

+ − Δ

∑ u (8-5o)

Page 136: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

135

2 2

16 , , , , , , ,1

2 3 2, , 4, , , ,

1 [ ( )2

( ) ]

k

y j y j x j x j x j y j y x jj

x j y j j x j y j y j j

D A h u c h u

c c c h h h n

ψ=

= − Δ + Δ − Δ

+ − Δ

∑ 2u (8-5p)

Again, for , we can either choose the coefficient in the x-ray error

expression or the coefficient in the y-ray error expression. Their numerical values may

differ for any single surface

3 5 6 12, , ,D D D D

j , but the summation in the final image space is the same. In

our treatment above, the coefficient in x-ray error expression was chosen because it

contains fewer terms.

8.3 Simplification of the results

We now have all of the primary wave aberration coefficients for general anamorphic

systems in terms of the paraxial marginal and chief ray trace data in the two associated

RSOS. Again, let us use the corresponding paraxial definitions from chapter 2.11 to

simplify them so that we can rewrite equations (8-5) in a form as similar to the Seidel

aberrations for RSOS as possible.

For , by comparing with equation (7-6a), we have 1D

2 2 3 3 41 , , , , , , , 3, ,

1

3 3 4, 3, ,

1

1 {[ ( )] ( ) }8

1 1 ( ) .8 8

k

x j x j x j x j x j x j x j j x j jj

k

Ix x j j x j jj

D A h u c h u c c h

S c c h n

=

=

= − Δ + Δ + − Δ

= − − − Δ

n (8-6a)

Notice that the additional aspheric term is exactly the same structure as the aspheric

contribution in the Seidel aberrations for x-RSOS.

For , by comparing with equation (7-6b), we immediately get 2D

Page 137: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

136

2 2 3 3 42 , , , , , , , 5, ,

1

3 3 4, 5, ,

1

1 {[ ( )] ( ) }8

1 1 ( ) .8 8

k

y j y j y j y j y j y j y j j y j jj

k

Iy y j j y j jj

D A h u c h u c c h

S c c h n

=

=

= − Δ + Δ + − Δ

= − − − Δ

n (8-6b)

Again, notice that the additional aspheric term is the same structure as the aspheric

contribution in the Seidel aberrations for y-RSOS.

For , we have 3D

2 2 2 3 2 23 , , , , , , , , 4, , ,

1 1

1 1( ) ( )4 4

k k

.x j x j y j y j y j x j x j y j j x j y j jj j

D A h u h c u c c c h h= =

= − Δ + Δ − − Δ∑ ∑ n (8-6c)

For , by comparison with equation (7-6d), we have 4D

2 24 , , , , , , , , , , , , ,

1

2 3 3 3, , 3, , ,

3 3 3, 3, , ,

1

1 [ ( 2 26

3( ) ]

1 1 ( ) .2 2

k

)x j x j x j x j x j x j x j x j x j x j x j x j x jj

x x j x j j x j x j j

k

IIx x j j x j x j jj

D A h u c h u h u u c h h u

u c c h h n

S c c h h n

ψ=

=

= − Δ + Δ + Δ + Δ

− Δ + − Δ

= − − − Δ

(8-6d)

For , we have 5D

5 , , , , , , ,

1

2 3 2, , 4, , , ,

1

1 ( )2

1 ( ) .2

k

,x j x j y j y j y j y j y j x jj

k

x j y j j x j y j y j jj

D A h u u h h c u

c c c h h h n

=

=

= − Δ + Δ

− − Δ

∑ (8-6e)

For , we have 6D

2 2

6 , , , , , ,1

2 2 3 2, , , 4, , , ,

1 [ ( )2

( )

k

x j x j y j y j y j x jj

]x y j x j y j j x j x j y j j

D A h u c h u

u c c c h h h n=

= − Δ + Δ

−Ψ Δ + − Δ

∑ (8-6f)

Page 138: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

137

2 2, , , , , , ,

1

2 3 2, , 4, , , ,

1

1 ( )2

1 ( )2

k

.

x j x j y j x j y j y j x jj

k

x j y j j x j x j y j jj

A h u A h c u

c c c h h h n

=

=

= − Δ + Δ

− − Δ

For , by comparison with equations (7-6g) through (7-6i), we obtain 7 8 9, ,D D D

2 27 , , , , , , , , , , , , ,

1

2 3 3 3, , 5, , ,

3 3 3, 5, , ,

1

1 [ ( 2 26

3( ) ]

1 1 ( ) .2 2

k

y j y j y j y j y j y j y j y j y j y j y j y j y jj

y y j y j j y j y j j

k

IIy y j j y j y j jj

D A h u c h u h u u c h h u

u c c h h n

S c c h h n

=

=

= − Δ + Δ + Δ + Δ

−Ψ Δ + − Δ

= − − − Δ

)

(8-6g)

2 28 , , , , , , , , , , , ,

1

3 3 2 2, , , 3, , ,

3 3 2 2, 3, , ,

1

1 [ ( 2 24

2 3( ) ]

1 3(3 ) ( ) .4 4

k

, )x j x j x j x j x j x j x j x j x j x j x j x j x jj

x x j x j x j j x j x j j

k

IIIx IVx x j j x j x j jj

D A h u c h u h u u c h h u

u u c c h h n

S S c c h h n

ψ=

=

= − Δ + Δ + Δ + Δ

− Δ + − Δ

= − + − − Δ

(8-6h)

2 29 , , , , , , , , , , , , ,

1

3 3 2 2, , , 5, , ,

3 3 2 2, 5, , ,

1

1 [ ( 2 24

2 3( ) ]

1 3(3 ) ( ) .4 4

k

y j y j y j y j y j y j y j y j y j y j y j y j y jj

y y j y j y j j y j y j j

k

IIIy IVy y j j y j y j jj

D A h u c h u h u u c h h u

u u c c h h n

S S c c h h n

ψ=

=

= − Δ + Δ + Δ + Δ

− Δ + − Δ

= − + − − Δ

)

(8-6i)

For , we have 10D

2 2 2 3 2 210 , , , , , , , , 4, , ,

1 1

1 1( ) ( )4 4

k k

.x j x j y j y j y j x j x j y j j x j y j jj j

D A h u h c u c c c h h= =

= − Δ + Δ − − Δ∑ ∑ n (8-6j)

For , we have 11D

2 2 2 3 2 211 , , , , , , , , 4, , ,

1 1

1 1( ) ( )4 4

k k

y j y j x j x j x j y j x j y j j x j y j jj j

D A h u h c u c c c h h n= =

= − Δ + Δ − − Δ∑ ∑ . (8-6k)

For , we have 12D

Page 139: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

138

12 , , , , , , , , , ,1

2 3, , 4, , , , ,

, , , , , , , , ,1

2 3, , 4, , , , ,

1

[ ( )

( ) ]

( )

( ) .

k

x j x j y j y j y j y j y j x j x y j y jj

x j y j j x j x j y j y j j

k

x j x j y j y j x j y j y j y j x jj

k

x j y j j x j x j y j y j jj

D A h u u c h h u u

c c c h h h h n

A h u u A h h c u

c c c h h h h n

ψ=

=

=

= − Δ + Δ − Δ

+ − Δ

= Δ + Δ

− − Δ

u

(8-6l)

For , by comparing with equations (7-6m) and (7-6n), we get 13 14,D D

2 2 2 3 3 313 , , , , , , , , 3, , ,

1

3 3 3, 3, , ,

1

1 [ ( ) ( )2

1 1 ( ) .2 2

k

]x j x j x j x j x j x j x x j x j j x j x j jj

k

Vx x j j x j x j jj

D A h u c h u u c c h h

S c c h h n

ψ=

=

= − Δ + Δ − Δ + − Δ

= − − − Δ

n (8-6m)

2 2 2 3 3 314 , , , , , , , , 5, , ,

1

3 3 3, 5, , ,

1

1 [ ( ) ( )2

1 1 ( ) .2 2

k

y j y j y j y j y j y j y y j y j j y j y j jj

k

Vy y j j y j y j jj

D A h u c h u u c c h h

S c c h h n

ψ=

=

= − Δ + Δ − Δ + − Δ

= − − − Δ

]n (8-6n)

For , we have 15D

2 2 215 , , , , , , ,

1

2 3 2, , 4, , , ,

2 2 2 3 2, , , , , , , , , 4, , , ,

1 1

1 [ ( )2

( ) ]

1 1( ) ( )2 2

k

x j x j y j y j y j x j x y jj

x j y j j x j x j y j j

k k

x j x j y j x j y j y j x j x j y j j x j x j y j jj j

D A h u c h u u

c c c h h h n

A h u A h c u c c c h h h n

ψ=

= =

= − Δ + Δ − Δ

+ − Δ

= − Δ + Δ − − Δ

∑ ∑

(8-6o)

For , we have 16D

2 2 216 , , , , , , ,

1

2 3 2, , 4, , , ,

2 2 2 3 2, , , , , , , , , 4, , , ,

1 1

1 [ ( )2

( ) ]

1 1( ) ( )2 2

k

y j y j x j x j x j y j y x jj

x j y j j x j y j y j j

k k

y j y j x j y j x j x j y j x j y j j x j y j y j jj j

D A h u c h u u

c c c h h h n

A h u A h c u c c c h h h n

ψ=

= =

= − Δ + Δ − Δ

+ − Δ

= − Δ + Δ − − Δ

∑ ∑

(8-6p)

Page 140: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

139

8.4 Summary

In this chapter, we have found the primary wave aberration coefficients for general

anamorphic systems in subgroups as:

Primary wave aberration coefficients associated with x-RSOS

3 3 41 , 3,

1

1 1 ( )8 8

k

Ix x j j x j jj

D S c c h=

, n= − − − Δ∑ ,

3 3 34 , 3,

1

1 1 ( )2 2

k

IIx x j j x j x j jj

D S c c h h=

= − − − Δ∑ , , n ,

3 3 2 28 ,

1

1 3(3 ) ( )4 4

k

IIIx IVx x j j x j x j jj

D S S c c h h=

= − + − − Δ∑ 3, , , n ,

3 3 313 , 3, , ,

1

1 1 ( )2 2

k

Vx x j j x j x j jj

D S c c h h=

= − − − Δ∑ n .

Primary wave aberration coefficients associated with y-RSOS

3 3 42 , 5,

1

1 1 ( )8 8

k

Iy y j j y j jj

D S c c h=

, n= − − − Δ∑ ,

3 3 37 , 5,

1

1 1 ( )2 2

k

IIy y j j y j y j jj

D S c c h h=

= − − − Δ∑ , , n ,

3 3 2 29 ,

1

1 3(3 ) ( )4 4

k

IIIy IVy y j j y j y j jj

D S S c c h h=

= − + − − Δ∑ 5, , , n ,

3 3 314 , 5, , ,

1

1 1 ( )2 2

k

Vy y j j y j y j jj

D S c c h h=

= − − − Δ∑ n .

Additional terms for skew rays

2 2 2 3 2 23 , , , , , , , , 4, ,

1 1

1 1( ) ( )4 4

k k

,x j x j y j y j y j x j x j y j j x j y j jj j

D A h u h c u c c c h h= =

= − Δ + Δ − − Δ∑ ∑ n ,

Page 141: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

140

2 3 25 , , , , , , , , , , 4, , , ,

1 1

1 1( ) ( )2 2

k k

x j x j y j y j y j y j y j x j x j y j j x j y j y j jj j

D A h u u h h c u c c c h h h= =

= − Δ + Δ − − Δ∑ ∑ n ,

2 2 2 3 26 , , , , , , , , , 4, , , ,

1 1

1 1( ) ( )2 2

k k

x j x j y j x j y j y j x j x j y j j x j x j y j jj j

D A h u A h c u c c c h h h n= =

= − Δ + Δ − − Δ∑ ∑ ,

2 2 2 3 2 210 , , , , , , , , 4, , ,

1 1

1 1( ) ( )4 4

k k

x j x j y j y j y j x j x j y j j x j y j jj j

D A h u h c u c c c h h= =

= − Δ + Δ − − Δ∑ ∑ n ,

2 2 2 3 2 211 , , , , , , , , 4, , ,

1 1

1 1( ) ( )4 4

k k

y j y j x j x j x j y j x j y j j x j y j jj j

D A h u h c u c c c h h n= =

= − Δ + Δ − − Δ∑ ∑ ,

2 312 , , , , , , , , , , , 4, , , , ,

1 1

( ) ( )k k

x j x j y j y j x j y j y j y j x j x j y j j x j x j y j y j jj j

D A h u u A h h c u c c c h h h h n= =

= Δ + Δ − −∑ ∑ Δ ,

2 2 2 3 215 , , , , , , , , , 4, , , ,

1 1

1 1( ) ( )2 2

k k

x j x j y j x j y j y j x j x j y j j x j x j y j jj j

D A h u A h c u c c c h h h= =

= − Δ + Δ − − Δ∑ ∑ n ,

2 2 2 3 216 , , , , , , , , , 4, , , ,

1 1

1 1( ) ( )2 2

k k

y j y j x j y j x j x j y j x j y j j x j y j y j jj j

D A h u A h c u c c c h h h= =

= − Δ + Δ − − Δ∑ ∑ n .

If we compare the results with chapter 7, we will find that the newly added terms

have similar structures as the aspheric terms of an RSOS.

For all the anamorphic primary aberration coefficient expressions derived from

chapters 5 through 8, we can not emphasize more on the fact that all parameters shown

up in these expressions are the paraxial marginal and chief rays’ tracing data in the two

associated RSOS, together with other first-order constants and definitions. Thus, for

anamorphic primary aberration calculation purpose, we only need to trace the four non-

skew marginal and chief rays, in the associated x-RSOS and y-RSOS. Hence, our results

are indeed similar to the Seidel aberrations of RSOS.

Page 142: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

141

CHAPTER 9

TESTING OF THE RESULTS

In chapters 5 through 8, we found both the primary ray aberration coefficients and

the primary wave aberration coefficients for the most common types of anamorphic

systems: from the simplest parallel cylindrical anamorphic systems to the most general

anamorphic systems made from general double curvature surfaces with the allowance of

four order aspheric departures. In this chapter, we will provide a testing scheme for the

primary aberration coefficient expressions developed.

Generally speaking, there are two common methods of testing the validity of a

theoretical result. The first one is to check the analytical development procedures. If none

of the steps in getting the final result can logically be disputed, the analytical result is

mathematically sound. The second method is to verify the results numerically. Nowadays,

it is very easy to setup different anamorphic systems using computer simulation software,

like ZEMAX or CODE V. The simulation software has the ability to trace real rays thus

providing the actual aberrations, and we can then compare the theoretical results

calculated from our expressions with the actual ray trace data and verify the validity of

our results.

Here we will make use of the ZEMAX program for the follows:

1) We will first assume that the analytical expressions we obtained are exact, and we will

use those expressions to calculate the primary wave aberration coefficients for any

arbitrary given anamorphic system;

Page 143: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

142

2) We will then calculate the same primary wave aberration coefficients for the given

anamorphic system using numerical real ray-tracing and data fitting;

3) Once we get both results, we will compare them and check the percentage errors

between the theoretical results and the real ray trace fitted results, thus verify the

validation of our analytical expressions.

This chapter presents the idea of data fitting in section 9.1, the detailed primary

aberrations coefficients data fitting steps in section 9.2, and an actual testing example in

section 9.3.

9.1 The idea of data fitting

From the general aberration theory described in chapter 3.3, we know the aberration

function can be expanded into a power series with respect to aperture and field

variables. The expansion expression is actually the aberration data fitting formula, which

tells that we can use the first-order, the third-order, the fifth order, and higher order terms

to represent the total aberration of a ray in an anamorphic system.

W

Since primary ray aberrations and primary wave aberrations are related by the

derivatives of the aperture variables, we can choose to either fit the real ray errors or the

real OPD error. Here we choose to fit the real ray errors since they are more

straightforward to calculate in ray tracing programs.

When a ray is traced through a given anamorphic system form a choosing object field,

as it meets the final ideal image plane k , we can easily obtain the coordinates ( ,k kξ η ) of

the meeting point using the ray tracing program.

Page 144: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

143

It is also easy to obtain the coordinates ( 0,k k 0ξ η ) of the ideal final image point for the

choosing object field by tracing the paraxial chief ray, which passes through the center of

the system stop.

Once we obtain these data from ray tracing, we can calculate the numerical value of

the x-component and y-component of the ray error in the final ideal image plane using

0k kδξ ξ ξ= − , (9-1a)

0k kδη η η= − . (9-1b)

Notice that equations (9-1a) and (9-1b) are total ray errors, which means that besides the

primary ray errors, the higher order ray errors are also included in these equations.

Theoretically, we can now use the first-order, the third-order, the fifth order and

higher order aberration terms to fit the data calculated in equations (9-1a) and (9-1b). But

this task is harder than we might have thought. In an anamorphic system, for the third-

order aberrations alone, we will have 16 terms already. Then how many terms will there

be for the fifth order, seventh order, ninth order and even higher orders? The answer

might be hundreds or even thousands. Thus it is not so practical to apply the data fitting

idea on equations (9-1) directly, and we need to go one step further.

In the deduction of this work, since we always use the ideal image point as our

reference, we know the first-order terms in the fitting expressions must always be zero.

We also know that, due to the higher order dependence of aperture and field, as we

scale both the system aperture and field down, the fifth and higher order aberration terms

will vanish very rapidly and will leave equations (9-1a) and (9-1b) with very little

contribution from the higher order aberration terms. Thus, under small aperture and field

Page 145: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

144

condition, we can treat equations (9-1) as if all contributions are from the third-order

aberration terms only.

From equations (3-7a) and (3-7b), we know the primary ray error fitting equations

are

3 2 2 2

1 3 4 5 6 8

2 3 210 12 13 15 ,

(4 2 3 2 2

2 )

2

/ ' ' ,x x y x x y x y x y x x

y x x y y x x y x k

D D D H D H D H D H

D H D H H D H D H H n u

δξ ρ ρ ρ ρ ρ ρ ρ

ρ ρ

= + + + + +

+ + + +

ρ

2

' ' .y

(9-2a)

3 2 2 2

2 3 5 6 7 9

2 3 211 12 14 16 ,

(4 2 2 3 2

2 ) /y x y y x x x y y y y

x y x y x y x y y k

D D D H D H D H D H

D H D H H D H D H H n u

δη ρ ρ ρ ρ ρ ρ ρ

ρ ρ

= + + + + +

+ + + +

ρ (9-2b)

From the discussion above, for any giving anamorphic imaging system, we can scale

down the system aperture and field so that we can treat the ray errors calculated from

equations (9-1) as if they are all primary ray errors. Thus we can use equations (9-2a)

and (9-2b) to fit them and obtain the primary aberration coefficients through with

high accuracy.

1D 16D

We know as we scale down the system aperture and field, the primary aberration

coefficients will be scaled down too, but it is not a concern since they are just linearly

scaled. What important to us is that as the aperture and field getting smaller, the higher

order aberration terms will be vanishing rapidly, thus the total ray errors are approaching

the primary ray errors rapidly. And the net result should be that the fitted aberration

coefficients are approaching the theoretical calculated primary aberration coefficients—if

the analytical expressions we developed in chapters 5 through 8 are accurate.

9.2 The detailed primary aberration coefficients data fitting steps

Page 146: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

145

Now we will describe the primary aberration coefficients data fitting steps in great

details. For any given anamorphic system, it is very easy to trace the four paraxial rays,

namely the x-marginal ray, the x-chief ray, the y-marginal ray, and the y-chief ray, in the

corresponding associated x-RSOS and y-RSOS, and find the corresponding paraxial

quantities, like ,' , ' ,x k yu u k used in equations (9-2), etc.

By examining equations (9-2a) and (9-2b) carefully, we can find the following

optimal steps in calculating the primary aberration coefficients, one by one.

1) By taking and(0,0)H = (1,0)ρ = , from equation (9-2a), we have

3

1 , 1

1 ,

4 / ' ' 4 / ' '( ' ' ) / 4.

,x x k x k

x k

D n u D n uD n u

δξ ρδξ

= =

⇒ = (9-3a)

For this specified field and aperture, we trace the real ray though the anamorphic system

and calculate δξ from equations (9-1) and then we can calculate the value of . 1D

2) Similarly, by taking (0,0)H = and (0,1)ρ = , from equation (9-2b), we have

(9-3b) 3

2 , 2

2 ,

4 / ' ' 4 / ' '

( ' ' ) / 4.y y k

y k

D n u D n u

D n u

δη ρ

δη

= =

⇒ =,y k

By applying exactly the same method as described in the calculation of , we can

calculate the value of .

1D

2D

3) By taking , (0,0)H = ( 2 / 2, 2 / 2)ρ = , we have

3 21 3 , 1 3(4 2 ) / ' ' (4 2 ) 2 / 4 ' ' ,x x y x k x kD D n u D D n uδξ ρ ρ ρ= + = + .

Since we already know from step 1), we can calculate 1D

3 ,2 ' ' 2x kD n u δξ= 1D− . (9-3c)

Page 147: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

146

4) By taking = (1,0), H ρ = (0,0) ,we have

313 , 13 ,/ ' ' / ' 'x x k x kD H n u D n uδξ = = ,

so we can calculate

13 ,' 'x kD n u δξ= . (9-3d)

5) By taking = (0,1), H ρ = (0,0), we have

, 314 , 14 ,/ ' ' / ' 'y y kD H n u D n uδη = = y k

so we can calculate

14 ,' ' y kD n u δη= . (9-3e)

6) By taking = (H 2 / 2 , 2 / 2 ), ρ = (0,0) , we have

3 213 15 , 13 15 ,( ) 2 / 4 / ' ' ( ) 2 / 4 ' 'x x y x k x kD H D H H n u D D n uδξ = + = + ,

3 214 16 , 14 16 ,( ) 2 / 4 / ' ' ( ) 2y x y y kD H D H H n u D D n uδη = + = + / 4 ' ' y k .

Since we already know , in steps 4) and 5), we can calculate 13D 14D

15 , 132 2 ' 'x kD n u δξ= D− , (9-3f)

16 , 142 2 ' ' y kD n u δη= D− . (9-3g)

7) By taking = (0, 1), H ρ = (1, 0), we have

3 21 10 , 1 10(4 2 ) / ' ' (4 2 ) / ' ' ,x y x x k x kD D H n u D D n uδξ ρ ρ= + = + ,

. 2 35 14 , 5 14( ) / ' ' ( )y x y y k y kD H D H n u D D n uδη ρ= + = + ,/ ' '

14

And since we know already, we can calculate 1 14,D D

5 ,' ' y kD n u Dδη= − , (9-3h)

Page 148: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

147

10 , 1( ' ' 4 ) / 2x kD n u Dδξ= − . (9-3i)

8) By taking = (1, 0), H ρ = (0, 1), we have

2 36 13 , 6 13( ) / ' ' ( ) ,/ ' 'x y x x k x kD H D H n u D D n uδξ ρ= + = + ,

. 3 22 11 , 2 11(4 2 ) / ' ' (4 2 ) / ' 'y x y y kD D H n u D D n uδη ρ ρ= + = + ,y k

3

13

And since we know already, we can calculate 2 1,D D

6 ,' 'x kD n u Dδξ= − , (9-3j)

11 , 2( ' ' 4 ) / 2y kD n u Dδη= − . (9-3k)

9) By taking = (H 2 / 2 , 2 / 2 ), ρ = (1, 0), we have

2 3 2

5 12 14 16

5 12 14 16 ,

( )

( 2 / 2 / 2 2 / 4 2 / 4) / ' ' .y x x y x y x y y k

y k

D H D H H D H D H H n u

D D D D n u

δη ρ ρ= + + +

= + + +

,/ ' '

And since we know already, we can calculate 5 14 16, ,D D D

12 , 5 14 162 ' ' 2 ( ) 2 / 2y kD n u D D Dδη= − − + . (9-3 l )

The calculations of the remaining 4 terms are not as simple as the

calculation of other terms because they are always coupled together. To separate them,

we need to make use of multiple field points.

4 7 8, , ,D D D D9

10) By taking = (1, 0), H ρ = (1, 0), we have

3 2 2 3

1 4 8 13

1 4 8 13 ,

(4 3 2 ) / ' '(4 3 2 ) / ' ' .

,x x x x x x x k

x k

D D H D H D H n uD D D D n u

δξ ρ ρ ρ= + + +

= + + + (9-4a)

Since we know already, equation (9-4a) is a linear equation of variables .

Now by taking = (0.5, 0),

1 13,D D 4 8,D D

H ρ = (1, 0), we have

Page 149: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

148

3 2 2 3

1 4 8 13

1 4 8 13 ,

(4 3 2 ) / ' '(4 3 / 2 / 2 /8) / ' ' .

,x x x x x x x k

x k

D D H D H D H n uD D D D n u

δξ ρ ρ ρ= + + +

= + + + (9-4b)

Equation (9-4b) is a linear equation of variables too, so by combining with

equation (9-4a), we can calculate .

4 ,D D8

4 8,D D

11) By taking = (0, 1), H ρ = (0, 1), we have

(9-5a) 3 2 2 3

2 7 9 14

2 7 9 14 ,

(4 3 2 ) / ' '

(4 3 2 ) / ' ' .y y y y y y

y k

D D H D H D H n u

D D D D n u

δη ρ ρ ρ= + + +

= + + +,y k

Since we know already, equation (9-5a) is a linear equation of variables .

Now by taking = (0, 0.5),

2 14,D D 7 9,D D

H ρ = (0, 1), we have

(9-5b) 3 2 2 3

2 7 9 14

2 7 9 14 ,

(4 3 2 ) / ' '

(4 3 / 2 / 2 /8) / ' 'y y y y y y

y k

D D H D H D H n u

D D D D n u

δη ρ ρ ρ= + + +

= + + +,y k

Equation (9-5b) is a linear equation of variables too, so by combining with

equation (9-5a), we can calculate .

7 9,D D

7 9,D D

Following the above detailed steps, we can find the fitted expansion coefficients

though using real ray tracing data from different field and aperture point, providing

that the aperture and field are small.

1D

16D

9.3 A testing example

From the discussion presented in sections 9.1 and 9.2, we know that if the two

conditions described below are satisfied, the analytical expressions under test must be

accurate:

Page 150: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

149

1) The primary aberration coefficients though calculated from the theoretical

expressions developed in chapters 5 through 8 are very close to the fitted though ,

for any corresponding anamorphic system type with small aperture and field;

1D 16D

1D 16D

2) Furthermore, as we gradually decrease the aperture and field, the fitted primary

aberration coefficients are approaching the theoretically calculated results.

In practice, the calculation of the primary aberration coefficients from the theory and

from the data fitting can be written into a ZEMAX macro and be applied onto different

anamorphic systems easily. The author of this work has tested many different anamorphic

systems using such a macro and proved that the expressions developed in chapters 5

through 8 are accurate.

Here, we will provide a simple testing example to illustrate the ideas. Consider an

anamorphic system in which two toroidal lenses together with a spherical lens, forms an

anamorphic image with an anamorphic ratio of 4:3. The effective focal length in the x-z

principal section is 40mm, and in the y-z principal section it is 30mm. Figure 9-1 below

illustrates the layouts in both principal sections. Notice that the system is uncorrected

with large aberrations. Table 9-1 lists the lens data. Figure 9-2 is the grid distortion map,

which shows the 4:3 image aspects.

Figure 9-1 Layout of a simple anamorphic system in the y-z (left) and x-z principal sections

Page 151: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

150

Table 9-1 Lens data

Figure 9-2 Grid distortion map

We will explore the effect of scaling down the system aperture and field while

keeping all other system parameters fixed.

In Table 9-2, the full system aperture is 10mm, and the half field of view (HFOV) is

10 degrees.

In Table 9-3, the system aperture is decreased to 4mm and the HFOV is 4 degrees.

In Table 9-4, the system aperture is further decreased to 1mm and the HFOV is 1

degree.

In each case, we will compare the fitted primary aberration coefficients with the

theoretic results calculated from the analytical aberration coefficient expressions.

Page 152: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

151

Table 9-2 Full aperture is 10mm, HFOV is 10 degree

From table 9-2, we see that for this system, even though the field and aperture are not

really small, the fitted data are actually not too far away from their corresponding

theoretical results, with a minimum accuracy of 94.9%.

Page 153: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

152

Table 9-3 Full Aperture is 4mm, HFOV is 4 degree

From table 9-3, we clearly see that as the field and aperture are getting smaller, the

fitted results are approaching the theoretical results very rapidly, with a minimum

accuracy of 99.18% now.

Page 154: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

153

Table 9-4 Full Aperture is 1mm, HFOV is 1 degree

From table 9-4, we see that for a small field and aperture, the fitted data can hardly

be distinguished from the theoretical results, with a minimum accuracy of 99.948% now.

Page 155: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

154

From this arbitrarily chosen example presented above, we indeed see that as the

aperture and field reduces in each instance, the theoretical results and the data fitted

results approach each other very rapidly and they will finally reach a level with extremely

high accuracy. This validates our theoretical expressions.

9.4 Summary

In this chapter, we provided a testing scheme for the expressions we had obtained in

chapters 5 through 8.

We applied the testing method described above to many kinds of arbitrary

anamorphic systems and the conclusions coming out are always similar. It is clear then

that the theoretical expressions we developed in chapters 5 through 8 must be exact.

Page 156: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

155

CHAPTER 10

DESIGN EXAMPLES

In the previous chapters, the complete monochromatic primary aberration theories for

most common types of anamorphic systems were presented. In this chapter, we will

present some design examples that illustrate the use and value of the theoretical results

developed in this work. The design examples given here will only cover a small part of

all of the possible anamorphic designs. However, they provide some insight into the

usefulness of the theoretical results.

In current practice, almost all anamorphic systems are designed using cylindrical

lenses because they are easy to manufacture, and as a result, they are cost-efficient.

However, since a cylindrical surface is a special case of toroidal surfaces, this chapter

will primarily use toroidal surfaces to illustrate the general design idea instead of being

limited to cylindrical anamorphic systems.

We will explore the design possibilities with increasing complexity. In section 10.1,

we will discuss the simplest anamorphic imaging system—an anamorphic singlet. In this

section, the basic ideas of anamorphic image formation will be presented.

In section 10.2, the idea of afocal attachments will be discussed. The simplest single

lens afocal attachment working with a RSOS landscape lens will be presented.

In section 10.3, we will make the renowned RSOS Cooke Triplet into its anamorphic

form.

In section 10.4, we will discuss the design idea of anamorphic field lens.

Page 157: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

156

In section 10.5, we will make the classic Double Gauss lens into its anamorphic form,

with the help of an anamorphic field lens. And finally, in section 10.6, we will explore

the design of an anamorphic fisheye lens.

As described in chapter 1, since there is no unique entrance and exit pupil in an

anamorphic system, in all the design examples presented in this chapter, "ray aiming" to

the real stop are always turned on and the system aperture is set to "float by stop size"

[29].

Together with the layouts in both principal sections, we will primarily use the spot

diagrams to assess system performance because they are generated via sampling all over

the aperture, rather than sampling two cross-sections of the aperture only, as in the ray

fan plot case. This kind of all aperture sampling is very important for an anamorphic

system due to the lacking of rotational symmetry.

Sometimes the ray fan plot, the OPD fan, and the modulation transfer function (MTF)

will also be utilized as system performance assessment tools even though they can only

partially represent the non-symmetrical system performance.

All examples shown in this chapter will be designed using the ZEMAX program.

However, the design ideas can be applied to any other lens design program as well.

10.1 An anamorphic singlet

From the discussion in chapter 1, we know that a single double curvature surface can

not form a unique image, and thus is not an imaging system. The simplest anamorphic

system we can imagine is an anamorphic singlet which has two double curvature surfaces.

Page 158: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

157

For an anamorphic singlet with the stop at the lens, there are only four independent

degrees of freedom that are effective in controlling aberrations and focal length. These

variables are the four lens curvatures, or alternatively, two powers and two bending

factors. To minimize longitudinal color, a low dispersion glass will be used. Lens

thickness is an ineffective variable, therefore a reasonably thin lens is assumed.

With four degrees of freedom, only four optical properties can be controlled. The two

powers are always used to fix the effective focal lengths in both principal sections thus

determining the anamorphic ratio.

Normally, we want to use the two elemental bending factors to control and ,

which are the spherical aberrations in the two principal sections.

1D 2D

But for the anamorphic system to form a unique image, we need a special first-order

condition which requires that the back focal lengths (BFL) in both principal sections

should equal each other so that the final image planes in both principal sections can

coincide. This image forming condition is also accomplished by proper bending in both

principal sections, thus it will make the two bending variables no longer independent,

which means we can not control and simultaneously. 1D 2D

However, we will have some flexibility here because the BFL requirement and the

spherical aberrations controlling are both done by lens bending. Thus the two bending

variables can be so selected to satisfy the BFL requirement and to achieve balanced

reasonably small spherical aberrations in both principal sections. By this way, even

though we do not have enough degrees of freedom to control and simultaneously, 1D 2D

Page 159: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

158

we will minimize their cumulative effect while satisfying the unique image forming

requirement.

From the discussion above, we see that for the four degrees of freedom, the best

result one can achieve is an anamorphic imaging system with specified anamorphic ratio,

with somewhat well balanced spherical aberrations in both principal sections.

In the example presented here, the anamorphic singlet lens with the stop at the lens

will be optimized to minimize the RMS spot size monochromatically, on axis only. The

glass is Schott BK7, which has low dispersion to reduce the color effects. The Object is at

infinity. The wavelengths are the visible F, d, and C lights. The reference wavelength is

the d light. Other specifications are as listed in Table 10-1 below.

Figure 10-1 below is the layout of the optimized lens in the two principal sections.

Notice that BFL in both principal sections must be exactly the same if we are to assure a

unique image is formed. Also notice that the off-axis ray bundles indicate a large amount

of field curvature in both principal sections.

Figure 10-2 is the on-axis ray fan and spot diagram. Notice that from the ray fan plot,

the ray errors are the same in both Sagittal and Tangential directions, thus the on-axis

spot should be circular if we were considering an RSOS case. But the actually on-axis

spot is in a star shape due to the non-RSOS nature of our system. This example clearly

illustrates why we will primarily use the spot diagram instead of ray fan (or OPD fan or

MTF) as the major system performance assessment tool in this chapter.

Also notice the fact that since lens bending is done in such a way that the spherical

aberrations in both principal sections are well balanced, the on-axis spot is quite uniform.

Page 160: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

159

Table 10-1 Specifications of the anamorphic singlet

Figure 10-1 Layout in the y-z (left) and x-z (right) principal sections

Figure 10-2 On-axis system performance

As we can see from the spot diagram, the image quality for this anamorphic singlet is

not very good even on-axis. For this slow system, the RMS spot size is on the scale of

Page 161: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

160

70 mμ . This is a consequence of insufficient variables to correct all three on-axis

spherical-type aberrations and the longitudinal color. 1 2, ,D D D3

The lens data and the remaining primary aberration coefficients (at a half field of 10

degrees) are shown in Table 10-2 below.

Table 10-2 Lens data and remaining primary aberration coefficients

From the primary aberration coefficients presented above, it is quite clear that the

various off-axis astigmatism and field curvature terms are limiting the off-axis

performance.

Notice that each toroidal surface presented above has curvatures which are different

in sign in each principal section. This kind of surface may be difficult to make, but they

serve well in illustrating the design ideas.

Page 162: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

161

Recall that in the classical landscape lens design, the stop location is used as an

effective variable to control the tangential field curvature. However, in this anamorphic

singlet design, since the field curvatures for either the principal sections or the skew rays

are not the same, varying the stop location can not correct the various field terms

simultaneously. Thus, the landscape lens design idea will not work effectively here.

10.2 An afocal anamorphic attachment

Ever since Chrétien published his afocal anamorphic attachment design patent in

1934, considerable research interest has focused on this group of design configuration. To

date, the majority of the existing anamorphic systems can be classified under the same

heading—an anamorphic afocal attachment combined with a standard optical imaging

system for spherical power [30-37].

There are several reasons for this phenomenon. One reason may be that most

anamorphic systems have their applications in Cinemascope, where anamorphic

attachments with different anamorphic ratios might be required.

Another reason lies in aberration controlling. In the early years of optics, people did

not have a full understanding of the aberration behavior in anamorphic systems. An

afocal attachment working in substantially collimated space with weak power will

introduce less aberration from the very beginning. Thus, this configuration can benefit the

aberration controlling process, as described by C. G. Wynne in his 1956 paper [13].

Afocal anamorphic attachments have another advantage. When pointing at an object

in infinite, the entering collimated beam will exit the anamorphic attachment lens as a

Page 163: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

162

substantially collimated beam having a different beam shape. This allows the anamorphic

attachment lens to be attached to an ordinary imaging lens of a camera without affecting

the position of the image plane. Thus, no refocusing of the camera is necessary when the

anamorphic attachment lens is applied or when the attachment is rotated relative to the

camera [38].

Some nice discussions on this common configuration can be found in G. H. Cook’s

book [17], R. Kingslake’s pattern [39], and other place [40].

In this section, we will present a simple yet interesting enough single lens afocal

anamorphic attachment design example to illustrate the idea.

We design the attachment as a single toroidal lens, using Schott BK7 glass. Again,

we have four independent degrees of freedom for the lens (two powers and two bending

factors). We start the design by separately bending the front and rear surfaces in each

principal section to make them afocal, respectively. This will consume two degrees of

design freedom, one for each section. The afocal property in each principal section is

achieved by requiring that the corresponding paraxial marginal ray exit angle be equal to

zero, for a collimated incident beam.

We also need to control the exit beam diameter in each section so that we can control

the required anamorphic ratio. This will consume two degrees of design freedom too,

one for each section. The exit beam diameter in each principal section is controlled by its

corresponding paraxial marginal ray exit height at the rear surface of the attachment.

From the discussion presented above, we see that all four design degrees of freedom

are utilized in order to control the first-order properties of the attachment. We do not have

Page 164: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

163

any degree of freedom left to control aberrations in the attachment design. However, a

weak powered afocal design itself will introduce very little aberration so long as the

surface curvatures are not too strong.

Normally the glass thickness is not an effective design variable, but in the afocal

design example described here, glass thickness will have a strong effect on the resulting

surface curvatures: For the same exit beam diameter, the thicker the glass, the less

curvature will be required for the surfaces. Thus, the glass thickness is actually of critical

importance for aberration controlling in this sense. But as the glass thickens, the weight

of the attachment will increase rapidly. Thus, under practical consideration, the glass

thickness is chosen to be 5mm here.

Figure 10-3 shows the attachment in both principal sections. Notice that the exiting

ray bundle in each section is collimated with different diameters.

Figure 10-3 Layout in the y-z (left) and x-z (right) principal sections

To make the system more practical, we choose the major spherical objective to be a

front stop landscape lens with a 20 degrees half field of view, and with an EFFL (the

effective focal length) of 50mm. The landscape lens is working at F/10. The glass is

again Schott BK7, and the glass thickness is chosen to be 3mm.

Page 165: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

164

In the combined system, the object distance is infinity. The stop is located on the

back surface of the toroidal lens. The wavelengths are the visible F, d, and C lights. The

reference wavelength is the d light. Other system specifications are listed in Table 10-3.

Table 10-3 Specifications of the anamorphic system

Figure 10-4 Layout in the y-z (left) and x-z (right) principal sections

Figure 10-5 Spot diagram

Page 166: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

165

Figure 10-6 Ray fan

Table 10-4 Lens data and remaining primary aberration coefficients

Page 167: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

166

Figure 10-4 above shows the system layout after the afocal toroidal lens is attached to

the landscape lens. Figure 10-5 is the spot diagram. Figure 10-6 is the ray fan. Table 10-4

is the lens data and the remaining primary aberration coefficients.

System distortion is less than 2.3% at maximum field. Since distortion does not

decrease the image sharpness, for a simple two-element anamorphic landscape lens, this

distortion level should be acceptable.

From the system layout, we see that the field curvatures in both principal sections are

not as severe as in the anamorphic singlet design presented in section 10.1. However,

from the spot diagram, it is clear that as the field size increases, various astigmatism and

field curvature terms will increase rapidly, and will thus limit the off-axis performance.

This issue can also be seen from the primary aberration coefficients listed above.

Therefore, this design example actually reveals a critically important aspect of

anamorphic system design: Because the system powers, together with the power

distribution for both principal sections and for the skew rays, are quite different in

anamorphic systems, it is most likely the various astigmatism and field curvature terms

will be the limiting off-axis aberrations for this kind of system.

Again, notice that each toroidal surface presented above has curvatures which are

different in sign in each principal section. This kind of surface may be difficult to make,

but they serve well in illustrating the design idea.

If we allow the attachment to be slightly focal and also allow some defocus of the

system, we can get a slightly better design result than the one presented above.

Page 168: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

167

10.3 An anamorphic Triplet design

In sections 10.1 and 10.2, the design examples presented are uncomplicated, and yet

they illustrate the anamorphic design ideas well. The biggest problem with single element

anamorphic systems is that the anamorphic ratio is controlled by the bending of only two

surfaces. For a low anamorphic ratio, such as 1.1 to 1.2, it may work fine. However, as

the anamorphic ratio increases, the curvature of each surface will soon becomes too

strong, thereby increasing the amount of aberrations it generates to an unacceptable level.

To achieve a higher anamorphic ratio and better image quality, we have to develop

more complex configurations so that the system can have greater degrees of freedom for

aberration controlling. In this section, we will expand our design ideas to the anamorphic

triplet design.

The RSOS Cooke Triplet is a very famous design form invented by H. D. Taylor in

1893.It consists of two positive singlet elements and one negative singlet element, all of

which can be thin. Two sizable airspaces separate the three elements. The negative

element is located in the middle about halfway between the positive elements, thus

maintaining a large amount of symmetry [41,42].

The RSOS Cooke Triplet is an optical configuration that has eight degrees of

freedom. The major variables are six lens surface curvatures and two inter-element air

spacing.

Stop shift is not a degree of freedom for the classical Cooke Triplet design because

the design is nearly symmetrical about the middle element, which makes aberration

Page 169: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

168

control much easier. To retain as much symmetry as possible, the stop is normally located

at the middle element.

The aberrations that need to be controlled for RSOS are the five monochromatic

third-order Seidel aberrations, together with the first-order longitudinal and lateral color.

Thus, the RSOS Cooke Triplet has just enough degrees of freedom available to correct all

first and third-order aberrations while obtaining the desired system focal length.

But for an anamorphic triplet design, we will not have enough degrees of freedom to

correct all sixteen anamorphic primary aberrations plus the chromatic aberrations. To be

more precise, we do not even have enough degrees of freedom to correct the primary

aberrations associated with the two principal sections.

Here is the reasoning: Suppose we say one of the principal sections has been fully

corrected. This corrected principal section will now fix both the two inter-element

spacing and the BFL. Thus, for the other principal section, we only have six surface

curvature variables available now. Unfortunately, for these six degrees of freedom, we

need to control BFL in this section to guarantee that a unique image is formed, and we

also need to control EFFL in this section to satisfy the required anamorphic ratio. Thus,

in general, we will have only four degrees of freedom left, which is simply not enough

for aberration control in the second principal section alone. This is said without

mentioning the skew aberrations yet! The difficulty seems insurmountable.

Due to the lack of variables in the anamorphic triplet design procedure, we must pay

close attention to the system performance balancing so that the aberrations for rays lying

Page 170: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

169

in the symmetry planes and the aberrations for skew rays will not adversely influence

each other too much.

It is also clear that due to the lack of variables, as the system departs further from

rotational symmetry (as the anamorphic ratio increases), the system performance will

generally decrease. In this section, we will briefly explore the effect of increasing

anamorphic ratio on system performance.

10.3.1 General methods for discovering an anamorphic beginning design

When designing a lens, the first step that involves the computer is entering the

starting design. In computer aided lens design, the starting point is of crucial importance

since if the starting point is far from a good solution, the merit function will soon fall into

a local minimum and stop, often leaving the designer with a less than optimal result.

Generally speaking, for any toroidal anamorphic system with a specified

configuration and anamorphic ratio, there are two methods for establishing a reasonably

good starting design. The first method can be described as follows:

1) Start from a well-corrected RSOS design with the specified configuration;

2) Set the BFL in both principal sections to the same value, which will guarantee that a

unique final image is formed;

3) Increase the EFFL target in one principal section by a small amount, and decrease the

EFFL target in the other principal section by the same amount. By doing so, the overall

system effective F/No is substantially fixed, yet the anamorphic ratio is slightly altered;

Page 171: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

170

4) Optimize the system using the default merit function to shrink the RMS spot size. At

this step, we may also consider adding some special operands to control specified

aberrations, such as distortion, etc;

5) Repeat steps 3 and 4 several iterations. Each time the EFFL targets in both principal

sections will be slightly changed in the same direction as in the previous steps, until the

specified anamorphic ratio is finally met. Once the starting design is achieved, the further

optimization process can begin.

The above iterative process has the effect of keeping us in the same general solution

region, avoiding radical departures from the general design form. By doing so, we can

find a reasonably good starting anamorphic design from a well-corrected RSOS, and

normally this starting design will not be too far away from a good final design.

The second method for discovering a good starting point for any toroidal anamorphic

design with a specified configuration and anamorphic ratio can be described by the

follows:

1) Since a well corrected anamorphic system must also be well corrected in the two

principal sections;

2) Thus we can design two separate RSOS with the same glass types, the same inter-

element separations, the same stop size and location, and the same back focal lengths.

What will differ in both RSOS are the surface curvatures and the effective focal lengths,

which controls the specified anamorphic ratio;

3) We correct both RSOS separately, then put them together to form an anamorphic

system made from toroidal surfaces;

Page 172: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

171

4) We then require the BFL in both principal sections be equal and start further

optimization on the integrated system.

By designing the two associated RSOS separately then putting them together, we can

find a reasonably good starting design with the specified anamorphic ratio, which is

normally not too far away from a desirable final design.

In practice, both methods work fine. For illustration purpose, in the anamorphic

Triplet design example presented in this section, we will make use of the first method to

find a starting design. In the anamorphic Double Gauss design example presented in

section 10.5 below, we will make use of the second method to find a starting design.

10.3.2 An anamorphic Triplet with an anamorphic ratio of 1.22

To design an anamorphic Triplet with an anamorphic ratio of 1.22, we begin with a

ZEMAX sample RSOS triplet design file. The starting specifications are as follows: The

EFFL is 50mm. The stop size is fixed at 7.732mm with the ray aiming function in use.

The stop is on the rear surface of the negative element. The full field size is 40 degrees.

The wavelengths are at 0.48, 0.55 and 0.65 microns. The reference wavelength is at 0.55

microns. For the two positive elements, the glass is Schott Sk16, and for the negative

element, the glass is Schott F2.

To find a reasonably good starting design with the specified anamorphic ratio, we

will employ the first method here to illustrate its usage.

We require the BFL in both principal sections be equal. We then increase the EFLX

target by 2mm and decrease the EFLY target by 2mm in the merit function. We also add

Page 173: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

172

several special operands into the merit function to control spherical aberration, axial color,

lateral color, and distortion associated with each principal section. Other aberrations are

controlled by shrinking the polychromatic spot size with the default merit function

operands.

We optimize the system using the above merit function and get an anamorphic

system with EFLX=52mm and EFLY=48mm. We repeat several times the same iterating

process with a small EFLX increases and EFLY decreases, until we finally reach the

desired anamorphic ratio with EFLX= 55mm and EFLY=45mm. Other specifications are

given in Table 10-5 below.

The intermediate solution for the anamorphic Triplet obtained from the above

iterating method is actually quite good in performance, and the final solution is a

refinement to it.

In the final optimization process, we will allow defocus at the image plane. The only

aberration type we will control using special operands are various distortions. The reason

lies in the fact that distortion does not decrease image quality, thus a shrinking spot size

may yield a solution with unacceptable distortion levels.

After some minor adjustments, we find a final solution by using a default merit

function which shrinks polychromatic spot size for all field positions and all wavelengths,

while continuing to correct EFLX, EFLY, BFL and distortion with special operands.

Figure 10-7 below is the layout of the final solution in both principal sections.

Figures 10-8 and 10-9 are the corresponding spot diagram and ran fan. Table 10-6 has the

lens data and the remaining primary aberration coefficients.

Page 174: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

173

Table 10-5 Specifications of the anamorphic triplet with anamorphic ratio 1.22

Figure 10-7 Layout in the y-z (left) and x-z (right) principal sections

Figure 10-8 Spot diagram

Page 175: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

174

Figure 10-9 Ray fan

Table 10-6 Lens data and remaining primary aberration coefficients

Page 176: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

175

From the spot diagram shown above, we can see that this anamorphic Triplet design

is acceptable with an RMS spot size of smaller than 47 mμ over the fields, and the image

quality is quite uniform over the field of view.

From the aberration coefficients list above, together with the ray fans, we see that the

defocus is not primarily used to balance on-axis spherical aberration. Since at bigger

fields, the various astigmatism and field curvature terms are the limiting aberrations,

ZEMAX actually utilized defocus to primarily balance off-axis field curvatures instead,

thus resulting in a relatively large on-axis spot.

10.3.3 Another anamorphic Triplet with an anamorphic ratio of 1.35

Now let us explore the effect on image quality of increasing the anamorphic ratio.

Following the same design procedures as described in section 10.3.2, we arrive at another

final design result with EFLX= 57.5mm and EFLY=42.5mm, and thus the anamorphic

ratio is now 1.35. All other specifications are listed in Table 10-7 below.

Figure 10-10 below is the layout of the final solution in both principal sections.

Figures 10-11 and 10-12 show the corresponding spot diagram and ray fan. Table 10-8 is

the lens data and the remaining primary aberration coefficients.

From the spot diagram, together with the remaining primary aberration coefficients

listed in Table 10-8, we see that as the anamorphic ratio increases, image quality is

deteriorating very rapidly, with a maximum RMS spot size 75 mμ now. And again, the

most problematic aberrations are the various astigmatism and field curvature terms. The

maximum system distortion is about 4.45%.

Page 177: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

176

Table 10-7 Specifications of the anamorphic triplet with anamorphic ratio 1.35

Figure 10-10 Layout in the y-z (left) and x-z (right) principal sections

Figure 10-11 Spot diagram

Page 178: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

177

Figure 10-12 Ray fan

Table 10-8 Lens data and remaining primary aberration coefficients

Page 179: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

178

If we increase the anamorphic ratio even more, the image quality will deteriorate so

rapidly that the resulting systems are barely usable. For an anamorphic Triplet design that

does not have enough degrees of freedom to control all primary aberrations, an

appropriate choice might be to limit the anamorphic ratio to less than 1.35.

10.4 An anamorphic field lens design

In the design examples presented in the above sections, the system powers together

with the power distributions are different for the two principal sections and for the skew

rays. Because of this, the various anamorphic astigmatism and field curvature terms are

normally the limiting off-axis aberrations. This is an inherent feature of anamorphic

system design.

However, this feature can lead to the idea of anamorphic field lens design, which is

quite useful in many cases. We will show the anamorphic field lens design idea via a real

design example.

Suppose we want to design a classical RSOS Cooke Triplet to take a picture of a

human thumb, which may be useful in security device. Also suppose the thumb is placed

50mm before the triplet lens. Since the thumb is a curved object with a different radius of

curvature in different principal sections, we can model it is as a toroidal object.

For an RSOS Cooke Triplet, we know its field curvature will be rotationally

symmetric, so we cannot map a toroidal object surface onto a flat image plane without the

help of one or more elements with anamorphic power inside the system.

Page 180: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

179

After some thinking, we come to the idea of introducing a toroidal field lens near the

image plane to provide different field curvatures in different principal sections.

We start from the same ZEMAX sample RSOS Triplet design used in section 10.3.

We scale the lens down so that the EFFL equals 15mm and the system will be working

with a 2/3 inch CCD. We optimize the Triplet design for an object distance of 50mm

from the first lens surface. The maximum half object height is 14mm, which matches the

size of the thumb. The working F number of the Triplet is 6.67. The wavelengths are 0.48,

0.55 and 0.65 microns. The reference wavelength is at 0.55 microns.

After obtaining a reasonable starting RSOS design, we now change the object to a

toroidal surface with mm and60xr = − 30yr = − mm, which are roughly the curvatures of

a human thumb. We then introduce a toroidal field lens right before the paraxial image

plane and let the computer program shrink the polychromatic spot size over all fields

using a default merit function. The glass of the toroidal field lens is Schott BK7.

Figure 10-13 below is the layout of the final solution. Figure 10-14 is the spot

diagram. Table 10-9 lists the lens data and the remaining primary aberration coefficients.

Figure 10-13 Layout in the y-z (left) and x-z (right) principal sections

Page 181: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

180

Figure 10-14 Spot diagram

Table 10-9 Lens data and remaining primary aberration coefficients

Page 182: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

181

As we can see from the spot diagram, this system is very well-corrected, with spot

sizes close to the airy disk, which is the diffraction limited case.

From the remaining primary aberration coefficients listed above, it is clear that the

anamorphic field lens indeed works really well in reducing the various astigmatism and

field curvature terms in the system.

10.5 An anamorphic Double Gauss design with anamorphic ratio 1.5

In section 10.3, we explored the anamorphic Triplet design. Due to the lack of design

degrees of freedom, we noticed that as the anamorphic ratio increases, the image quality

deteriorates rapidly. Thus, for an anamorphic Triplet design, we should limit ourselves to

a relatively small anamorphic ratio. We also found that the various astigmatism and field

curvature terms will most likely be the limiting off-axis aberrations, which is an inherent

aberration feature for anamorphic image systems due to different powers and power

distributions in different principal sections.

In section 10.4, we introduced the idea of adding an anamorphic field lens to achieve

a flat field in both principal sections. Even tough the example presented in section 10.4 is

an RSOS system with a toroidal object surface, the idea is quite revealing and should be

applicable to general anamorphic imaging system design.

To achieve an anamorphic design with a higher anamorphic ratio and better image

quality, we need to have more design degrees of freedom, and we also need a way to

control the inherent field curvature terms. We will now combine the design ideas

Page 183: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

182

presented in the previous sections and extend our design example to the anamorphic

Double Gauss design with a field lens.

The RSOS Double Gauss is a very famous design form which was originally used for

wide angle lenses. It was originally constructed by placing two Gauss type achromatic

doublets back-to-back symmetrically about a stop. Later on a cemented buried surface

was added inside each negative meniscus, with the glasses on each side of the buried

surface having similar indices but different dispersions. Their purpose was to simulate a

glass with greater dispersion than any available. Finally, the system was made slightly

unsymmetrical about the stop and the ultimate design configuration was achieved (G. H.

Smith, 1998).

Now let us design an anamorphic Double Gauss with a moderate anamorphic ratio of

1.5. We will start from a scaled ZEMAX sample RSOS Double Gauss design. The

starting specifics are as follows: The EFFL is 50mm. The stop size is fixed at 10mm with

ray aiming in use. The stop is in the middle air space between the front and rear lens

groups. The full field size is 28 degrees. The wavelengths are the visible F, d, and C light.

For the two positive elements, the glasses used are Schott Sk2 and Sk16. The glass for the

negative element is Schott F5.

To start the anamorphic design, we must achieve a reasonably good starting point

with the specified configuration and the specified anamorphic ratio. As described in

section 10.3.2, generally speaking, there are two methods of approaching a good starting

design. In the anamorphic Triplet design example presented in section 10.3, we applied

the first approach. In this section, we will apply the second approach to illustrate its usage.

Page 184: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

183

Starting from the ZEMAX sample file, we design two RSOS Double Gauss lenses

with the same BFL of 30mm. The glasses, the inter-element separations, the stop size and

location are all the same. The things different in both systems are the surface curvatures

and the effective focal lengths, which control the anamorphic ratio. We design the first

RSOS with EFFL=50mm and the second RSOS with EFFL=75mm, thus the anamorphic

ratio is 1.5.

We optimize each RSOS separately then combine them to form an anamorphic

system made from toroidal surfaces. In this way, we find a reasonably good starting point

with the specified anamorphic ratio of 1.5.

Table 10-10 below is the design specifications. Figure 10-15 shows the starting

configuration in both principal sections. Figure 10-16 shows the spot diagram at this

stage.

Table 10-10 Specifications of the anamorphic Double Gauss

Figure 10-15 Layout of the initial design in the y-z (left) and x-z (right) principal sections

Page 185: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

184

Figure 10-16 Spot diagram for the initial design

From the system layout and spot diagram, it is clear that this initial design is

acceptable and is capable of serving as the starting point.

Now we require the BFL in both principal sections always be equal to ensure a

unique final image is formed. We then add some special operands into the default merit

function while shrinking the polychromatic spot size across the field. The special

operands are used to control all three spherical type aberrations and all coma type

aberrations. We will not specially control the various field curvature terms at this stage

because we will add a field lens to control them later.

After this first stage optimization, the spot diagram together with the remaining

primary aberrations is shown in Figure 10-17 below.

Page 186: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

185

Figure 10-17 Spot diagram and aberrations coefficients after the first stage optimization

From the spot diagram and the primary aberration coefficients table, we see the

various field Curvature terms are now indeed the dominant aberrations (ignore distortions

because they do not decrease image sharpness). Thus, it is the right time for the field lens

to be added.

Page 187: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

186

The field lens is added right in front of the paraxial focus plane with a center

thickness of 1mm and a glass type of Schott BK7. The first surface of the field lens is

toroidal and the second surface is flat. We get the initial radii of curvature for the toroidal

field lens by manually adjusting its radii while examining the spot diagram, and then

setting the surface curvatures free to vary.

After adding the field lens, we are approaching the final design. At this stage, image

plane defocus is set as a variable, all special operands are removed, and we let the

program shrink the polychromatic spot size across the field using a default merit function.

Of course, the operands for EFLX and EFLY and the operands for controlling BFL are

untouched. After the optimization and some manual refinement, we get a final design

with a total of 9 toroidal surfaces used.

Figure 10-18 below is the layout of the final solution. Figures 10-19 and 20 reflect

the system spot diagram and OPD fan. The lens data and the remaining primary

aberrations are shown in Table 10-11.

Figure 10-18 Layout of the anamorphic Double Gauss design

Page 188: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

187

Figure 10-19 Spot diagram

Figure 10-20 OPD fan

Page 189: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

188

Table 10-11 Lens data and remaining primary aberration coefficients

From the aberration coefficients list above, it is clear that the various field curvature

terms are controlled. System maximum distortion is controlled for less than 2.6%.

From the spot diagram and OPD fan, we can see clearly that the image quality of this

moderate anamorphic ratio Double Gauss design is very good with an RMS spot size of

less than 29 mμ across the field.

10.6 An anamorphic fisheye lens design with anamorphic ratio 3:4

In this section, we will explore the design of an anamorphic fisheye lens, which will

be working over a 140 degree full field of view. The anamorphic ratio will be 3:4.

Page 190: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

189

First of all, let us find an RSOS fisheye design as the beginning point. Starting from

the classical wide-angle Double Gauss configuration, we form the first and second

elements into meniscus shapes, which are common in fisheye type lenses. We also

gradually increase the front group elements separation during the optimization process so

that the incident ray height can be gradually decreased before it reaches the system stop.

After some trials, we arrived at a modified wide-angle Double Gauss configuration with

separated front elements.

We then play with the glass types and gradually increase the half field of view to 70

degrees. After some detailed adjustments, we arrive at a RSOS fisheye design as shown

in Figure 10-21 below, which will serve as our starting RSOS design.

Figure 10-21 A RSOS Double Gauss type fish-eye Lens

This is a modified Double Gauss configuration with separated front elements. The

EFFL is 50mm. The system is fixed at F/5 with ray aiming applied. The stop is in the

middle air space between the front and rear lens groups. The full field size is 140 degrees.

Page 191: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

190

The wavelengths are the visible F, d and C lights. In the above layout, from left to right,

the glasses used are Schott N-Sk14, Lak11, Sf1, N-Sk11, Sf57 and Laf22a.

10.6.1 An anamorphic fisheye design without field lens

From the starting RSOS fisheye design, utilizing the first design method described in

section 10.3, we gradually altered the EFLX and EFLY target while using the default

merit function to shrink the polychromatic spot size across the field. After several

iterating steps, we find a reasonably good starting anamorphic fisheye design, with the

specified anamorphic ratio of 3:4. The design specifications at this stage are listed in

Table 10-12 below.

Table 10-12 Initial design specifications of the anamorphic Double Gauss

Figure 10-22 below shows the initial anamorphic fisheye design in both principal

sections. Figure 10-23 shows the grid distortion at this stage.

Figure 10-22 Layout in the y-z (left) and x-z (right) principal sections

Page 192: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

191

Figure 10-23 Grid distortion for the initial anamorphic fisheye design

An examination of the grid distortion map reveals the image field size is too large,

thus we must scale the lens down so that the image can fit a standard 35 mm film format.

After the scaling, defocus is turned on as a variable, all special operands other than

EFLX, EFLY, and BFL are removed from the default merit function, and we let the

program shrink the polychromatic spot size across the field. After some refinement trials,

we achieve a final design. The final design specifications are listed in Table 10-13 below.

Figure 10-24 illustrates the final anamorphic fisheye design in both principal sections.

Table 10-14 lists the lens data. Figure 10-25 shows the grid distortion. Figures 10-26 and

27 represent the spot diagram and ray fan.

Table 10-13 Final design specifications of the anamorphic Double Gauss

Page 193: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

192

Figure 10-24 Layout of the final design in the y-z (left) and x-z (right) principal sections

Table 10-14 Lens data of the final design

Figure 10-25 Grid distortion map of the final design

Page 194: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

193

Figure 10-26 Spot diagram of the final design

Figure 10-27 Ray fan of the final design

Page 195: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

194

From the spot diagram, we see that system performance is very good with an RMS

spot size of less than 31 mμ over all fields. The maximum system distortion is 36.7%,

which is at the normal distortion level for a fisheye lens.

10.6.2 An anamorphic fisheye design with field lens

From the discussion presented in previous sections, we know that by adding a field

lens to reduce the effect of various field curvature terms, we can normally achieve a

design with better system performance.

Now let us add a cylindrical field lens near the final image plane. The glass type of

the cylindrical field lens is again Schott Bk7. We also introduce a little bit of vignetting

by limiting the semi-diameter of the surface right after the system stop. After some

detailed refinement, we achieve a final design. The specifications of the final design are

listed in Table 10-15 below.

Figure 10-28 below shows the layout of the final design in both principal sections.

Table 10-16 shows the lens data. Figure 10-29 shows the vignetting plot. Figure 10-30

shows the spot diagram. Figure 10-31 shows the ray fan.

Table 10-15 Specifications of the anamorphic Double Gauss with field lens

Page 196: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

195

Figure 10-28 Layout of the final design in the y-z (left) and x-z (right) principal sections

Table 10-16 Lens data of the final design with field lens

Figure 10-29 Vignetting plot

Page 197: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

196

Figure 10-30 Spot diagram

Figure 10-31 Ray Fan

Page 198: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

197

Examining the spot diagram reveals that system performance is much improved with

a new RMS spot size across the field less than 20 mμ . The vignetting we introduced is

very small with a relative geometrical transmission at the edge of the field larger than

86%. The maximum system distortion is now 39.2%.

10.7 Summary

In this chapter, a sampling of some different anamorphic designs has been presented.

We started with the simplest anamorphic singlet design and finished with a complex

anamorphic fisheye design. We introduced the important methods for approaching a

starting design, and we explained the limiting aberrations in each configuration and how

they are controlled.

From the examples presented in this chapter, together with the theoretical structures

presented in previous chapters, the reader of this work should have a better understanding

of anamorphic imaging system design.

Page 199: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

198

CHAPTER 11

CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK

11.1 Conclusions

In this work, a theoretical structure has been developed to describe the aberrations of

anamorphic optical systems that can be considered a generalization of the aberration

theory of rotationally symmetric optical system (RSOS).

In chapter 2, we found that we can think of a paraxial anamorphic system as two

RSOS, each associated with one symmetry plane. Thus all known results for the two

RSOS can be applied to the anamorphic system directly. We also found that there are

only two independent paraxial skew rays in an anamorphic system, but we prefer using

the four non-skew marginal and chief rays, traced in the two associated RSOS, to fully

specify the system. We found that by using these four paraxial rays, we can get all

paraxial quantities associated with the anamorphic system.

In chapter 4, by applying the generalized Aldis idea onto anamorphic systems, we

built up the anamorphic total ray aberration equations. We then reduced these equations

into the anamorphic primary ray aberration equations. We then wrote all parameters in

the anamorphic primary ray aberration equations in terms of the ray-tracing data of the

four non-skew paraxial rays in the two associated RSOS, together with object and stop

variables. By these steps, we built up a general method of deriving the primary aberration

coefficients for any anamorphic system types.

Page 200: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

199

In chapters 6, the monochromatic primary aberration coefficient expressions for

cross cylindrical anamorphic systems were found by applying the method we developed.

In chapters 7, the monochromatic primary aberration coefficient expressions for

toroidal anamorphic systems were found by applying the method we developed.

In chapters 8, the monochromatic primary aberration coefficient expressions for

general anamorphic systems were found by applying the method we developed.

All results listed above are novel and can not be found in existing literatures. Thus

our work greatly expanded the scope of current anamorphic imaging systems research.

11.2 Suggestions of future work

In this work, the ground work (first-order optics), the relevant concepts and the

primary aberration theory for understanding the imagery of the most common types of

anamorphic systems have been given. But of course, there is room for future work.

For example, the chromatic variation of the aberrations needs to be investigated. The

thin lens contributions, and the stop shift formulae for anamorphic systems need to be

derived, and the higher order aberrations need to be explored, etc.

More importantly, we need to apply the theoretical results to all kinds of actual

anamorphic designs so that the design experiences can be accumulated, as we did in the

past 150 years since Seidel presented the Seidel aberrations for RSOS.

Serious applications of the results presented in this work onto different anamorphic

system designs, together with other theoretical research should be able to further expand

the research in this field.

Page 201: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

200

APPENDIX A

THE TYPICAL SHAPE OF ANAMORPHIC PRIMARY WAVE ABERRATIONS

4 4

1 1 cosxW D D 4ρ ρ θ= ⋅ = ⋅ 4 42 2 sinyW D D 4ρ ρ θ= ⋅ = ⋅

2 2 4 2 23 3 sin cosx yW D Dρ ρ ρ θ= ⋅ = ⋅ θ 3 3 3

4 4 cosx x xW D H D Hρ ρ θ= ⋅ = ⋅

2 35 5 sin cosy x y yW D H D H 2ρ ρ ρ θ= ⋅ = ⋅ 2 3 2

6 6 sin cosx x y xW D H D Hθ

ρ ρ ρ θ θ= ⋅ = ⋅

Page 202: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

201

3 37 7 siny y yW D H D H 3ρ ρ θ= ⋅ = ⋅ 2 2 2 2 2

8 8 cosx x xW D H D Hρ ρ θ= ⋅ = ⋅

2 2 2 2 29 9 siny y yW D H D Hρ ρ θ= ⋅ = ⋅ 2 2 2 2 2

10 10 cosy x yW D H D Hρ ρ θ= ⋅ = ⋅

2 2 2 2 211 11 sinx y xW D H D Hρ ρ θ= ⋅ = ⋅ 12

212 sin cos

x y x y

x y

W D H H

D H H

ρ ρ

ρ θ θ

= ⋅

= ⋅

Page 203: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

202

3 313 13 cosx x xW D H D Hρ ρ θ= ⋅ = ⋅ 3 3

14 14 siny y yW D H D Hρ ρ θ= ⋅ = ⋅

2 215 15 cosx y x x yW D H H D H Hρ ρ θ= ⋅ = ⋅ 2 2

16 16 sinx y y x yW D H H D H Hρ ρ θ= ⋅ = ⋅

Notice that even though are of the same shape in aperture dependences, but due to their different field dependences, they are different aberration types. The same is true for .

13 15,D D

14 16,D D

Page 204: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

203

REFERENCES

[1] H. A. Buchdahl, "An introduction to Hamiltonian Optics", Cambridge (1970). [2] W. T. Welford, "Aberrations of Optical System", Adam Hilger Ltd (1986). [3] P. J. Sands, "Aberration Coefficients of Double-Plane-Symmetric Systems", J. Opt. Soc. Am. 63 (4). 425-430 (1973). [4] J. M. Sasian, "How to approach the design of a bilateral symmetric optical system", Opt. Eng. 33, 2045–2061 (1994). [5] P. J. Sands, "Aberrations Coefficients of Plane Symmetric Systems", J. Opt. Soc. Am. 62 (10), 1211-1220 (1972). [6] US Patent 5,671,093 (1997). [7] US Patent 7,085,066 B2 (2006). [8] S. Thibault "Distortion Control Offers Optical System Design a New Degree of Freedom", Photonics Spectra May 2005, pp. 80-82. [9] Rolland, Jannick P.; Rapaport, Alexandra; Krueger, Myron W. "Design of an anamorphic fish-eye lens", Proc. SPIE Vol. 3482, p. 274-277, International Optical Design Conference 1998. [10] Von Rohr in "The Formation of Images in Optical Instruments", London H. M. Stationary Office, 196-211 (1920). [11] G. J. Burch "Some Uses of Cylindrical Lens-Systems, Including Rotation of Images" Proceedings of the Royal Society of London, Vol. 73. (1904), pp. 281-286. [12] H. Chretien, US patent 1,962,892 "Anamorphotic lens system and method of making the same" (1929). [13] C. G. Wynne, "The Primary Aberrations of Anamorphic Lens Systems", Proc. Phys. Soc. Lond. 67b, 529-537 (1954). [14] J. C. Bourfoot, "Third-Order Aberrations of ‘Doubly Symmetric’ Systems", Proc. Phys. Soc. Lond. 67b, 523-528 (1954). [15] H. Kohler, "Zur Abbildungstheorie anamorphotischer Systeme",Optic 13, 145-157 (1956).

Page 205: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

204

[16] K. Bruder, "Die Bildfehler Dritter Ordnung in anamorphotischen Systemen", Optik 17, 663-670 (1960). [17] G. H.Cook, "Anamorphotic Systems." In Applied Optics and Optical Engineering, Academic Press, R. Kingslake Ed. 3, 127-132(1965). [18] G. G. Slyusarev, "Aberration and Optical Design Theory", Adam Hilger (1984). [19] E. Abbe, as cited by von Rohr in "The Formation of Images in Optical Instruments", London H. M. Stationary Office (1920). [20] R. Barakat and A. Houston. "The Aberrations on Non-Rotationally Symmetric Systems and their Diffraction Effects", Optica Acta, 13 (1), 1-30 (1966). [21] T. A. Kuz’mina, "Third-Order Aberrrations of Optical Systems with tow Symmetry Planes", Sov. J. Opt. Technol. 41, 8 (1974). [22] A. Cox, "A System of Optical Design ", by London, GB The Focal Press, (1964). [23] J. M. Sasian, "Double-curvature surfaces in mirror system design", Opt. Eng., Vol. 36, No. 1, p. 183 – 188 (1997). [24] E. Wandersieb, "Abbe's geometrical theory of the formation of optical images," in The formation of images in optical instruments, Von Rohr editor, H.M. Stationary Office, London, 83-124, (1920). [25] R. R. Shannon, "The Art and Science of Optical Design", Cambridge University Press (1997). [26] E. Hecht, "Optics", 4th Edition, Addison Wesley (2001). [27] M. Born and E. Wolf, "Principles of Optics", 7th Edition, Cambridge (2001). [28] H. H. Hopkins, "Wave Theory of Aberrations ", New York: Oxford Univ. Press, (1950). [29] ZEMAX user guide, July 2002. [30] US Patent 2,731,883 (1956). [31] US Patent 2,821,110 (1958). [32] US Patent 2,956,475 (1960).

Page 206: ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMSarizona.openrepository.com/arizona/bitstream/10150/195267/1/azu... · ABERRATIONS OF ANAMORPHIC OPTICAL SYSTEMS By Sheng Yuan A Dissertation

205

[33] US Patent 3,002,427 (1961). [34] US Patent 3,517,984 (1968). [35] US Patent 3,924,933 (1975). [36] US Patent 3,990,785 (1976). [37] US Patent 4,362,366 (1982). [38] US Patent 6,072,636 (2000). [39] US Patent 2,933,017 (1960). [40] US Patent 6,310,731 B1 (2001). [41] G. H. Smith, "Practical Computer-Aided Lens Design." Willamnn-Bell, INC (1998). [42] W. J. Smith, "Modern Lens Design." McGraw-Hill (2005).